Here is a list of all documented files with brief descriptions:

common
unicode
appendable.h	C++ API: Appendable class: Sink for Unicode code points and 16-bit code units (char16_ts)
brkiter.h	C++ API: Break Iterator
bytestream.h	C++ API: Interface for writing bytes, and implementation classes
bytestrie.h	C++ API: Trie for mapping byte sequences to integer values
bytestriebuilder.h	C++ API: Builder for icu::BytesTrie
caniter.h	C++ API: Canonical Iterator
casemap.h	C++ API: Low-level C++ case mapping functions
char16ptr.h	C++ API: char16_t pointer wrappers with implicit conversion from bit-compatible raw pointer types
chariter.h	C++ API: Character Iterator
dbbi.h	C++ API: Dictionary Based Break Iterator
docmain.h	(Non API- contains Doxygen definitions)
dtintrv.h	C++ API: Date Interval data type
edits.h	C++ API: C++ class Edits for low-level string transformations on styled text
enumset.h	C++: internal template EnumSet<>
errorcode.h	C++ API: ErrorCode class intended to make it easier to use ICU C and C++ APIs from C++ user code
filteredbrk.h	C++ API: FilteredBreakIteratorBuilder
icudataver.h	C API: access to ICU Data Version number
icuplug.h	C API: ICU Plugin API
idna.h	C++ API: Internationalizing Domain Names in Applications (IDNA)
localebuilder.h	C++ API: Builder API for Locale
localematcher.h	C++ API: Locale matcher: User's desired locales vs
localpointer.h	C++ API: "Smart pointers" for use with and in ICU4C C++ code
locdspnm.h	C++ API: Provides display names of Locale and its components
locid.h	C++ API: Locale ID object
messagepattern.h	C++ API: MessagePattern class: Parses and represents ICU MessageFormat patterns
normalizer2.h	C++ API: New API for Unicode Normalization
normlzr.h	C++ API: Unicode Normalization
parseerr.h	C API: Parse Error Information
parsepos.h	C++ API: Canonical Iterator
platform.h	Basic types for the platform
ptypes.h	C API: Definitions of integer types of various widths
putil.h	C API: Platform Utilities
rbbi.h	C++ API: Rule Based Break Iterator
rep.h	C++ API: Replaceable String
resbund.h	C++ API: Resource Bundle
schriter.h	C++ API: String Character Iterator
simpleformatter.h	C++ API: Simple formatter, minimal subset of MessageFormat
std_string.h	C++ API: Central ICU header for including the C++ standard <string> header and for related definitions
strenum.h	C++ API: String Enumeration
stringoptions.h	C API: Bit set option bit constants for various string and character processing functions
stringpiece.h	C++ API: StringPiece: Read-only byte string wrapper class
stringtriebuilder.h	C++ API: Builder API for trie builders
symtable.h	C++ API: An interface that defines both lookup protocol and parsing of symbolic names
ubidi.h	C API: Bidi algorithm
ubiditransform.h	C API: Bidi Transformations
ubrk.h	C API: BreakIterator
ucasemap.h	C API: Unicode case mapping functions using a UCaseMap service object
ucat.h	C API: Message Catalog Wrappers
uchar.h	C API: Unicode Properties
ucharstrie.h	C++ API: Trie for mapping Unicode strings (or 16-bit-unit sequences) to integer values
ucharstriebuilder.h	C++ API: Builder for icu::UCharsTrie
uchriter.h	C++ API: char16_t Character Iterator
uclean.h	C API: Initialize and clean up ICU
ucnv.h	C API: Character conversion
ucnv_cb.h	C API: UConverter functions to aid the writers of callbacks
ucnv_err.h	C API: UConverter predefined error callbacks
ucnvsel.h	C API: Encoding/charset encoding selector
uconfig.h	User-configurable settings
ucpmap.h	C API: This file defines an abstract map from Unicode code points to integer values
ucptrie.h	C API: This file defines an immutable Unicode code point trie
ucurr.h	C API: Encapsulates information about a currency
udata.h	C API: Data loading interface
udisplaycontext.h	C API: Display context types (enum values)
uenum.h	C API: String Enumeration
uidna.h	C API: Internationalizing Domain Names in Applications (IDNA)
uiter.h	C API: Unicode Character Iteration
uldnames.h	C API: Provides display names of Locale ids and their components
uloc.h	C API: Locale ID functionality similar to C++ class Locale
ulocale.h	C API: Locale ID functionality similar to C++ class Locale
ulocbuilder.h	C API: Builder API for Locale
umachine.h	Basic types and constants for UTF
umisc.h	C API: Miscellaneous definitions
umutablecptrie.h	C API: This file defines a mutable Unicode code point trie
unifilt.h	C++ API: Unicode Filter
unifunct.h	C++ API: Unicode Functor
unimatch.h	C++ API: Unicode Matcher
uniset.h	C++ API: Unicode Set
unistr.h	C++ API: Unicode String
unorm.h	C API: Unicode Normalization
unorm2.h	C API: New API for Unicode Normalization
uobject.h	C++ API: Common ICU base class UObject
urep.h	C API: Callbacks for UReplaceable
ures.h	C API: Resource Bundle
uscript.h	C API: Unicode Script Information
uset.h	C API: Unicode Set
usetiter.h	C++ API: UnicodeSetIterator iterates over the contents of a UnicodeSet
ushape.h	C API: Arabic shaping
usprep.h	C API: Implements the StringPrep algorithm
ustring.h	C API: Unicode string handling functions
ustringtrie.h	C API: Helper definitions for dictionary trie APIs
utext.h	C API: Abstract Unicode Text API
utf.h	C API: Code point macros
utf16.h	C API: 16-bit Unicode handling macros
utf32.h	C API: UTF-32 macros
utf8.h	C API: 8-bit Unicode handling macros
utf_old.h	C API: Deprecated macros for Unicode string handling
utfiterator.h	C++ header-only API: C++ iterators over Unicode strings (=UTF-8/16/32 if well-formed)
utfstring.h	C++ header-only API: C++ string helper functions
utrace.h	C API: Definitions for ICU tracing/logging
utypes.h	Basic definitions for ICU, for both C and C++ APIs
uvernum.h	C API: definitions of ICU version numbers
uversion.h	C API: API for accessing ICU version numbers
i18n
unicode
alphaindex.h	C++ API: Index Characters
basictz.h	C++ API: ICU TimeZone base class
calendar.h	C++ API: Calendar object
choicfmt.h	C++ API: Choice Format
coleitr.h	C++ API: Collation Element Iterator
coll.h	C++ API: Collation Service
compactdecimalformat.h	C++ API: Compatibility APIs for compact decimal number formatting
curramt.h	C++ API: Currency Amount Object
currpinf.h	C++ API: Currency Plural Information used by Decimal Format
currunit.h	C++ API: Currency Unit Information
datefmt.h	C++ API: Abstract class for converting dates
dcfmtsym.h	C++ API: Symbols for formatting numbers
decimfmt.h	C++ API: Compatibility APIs for decimal formatting
displayoptions.h	C++ API: Display options class
dtfmtsym.h	C++ API: Symbols for formatting dates
dtitvfmt.h	C++ API: Format and parse date interval in a language-independent manner
dtitvinf.h	C++ API: Date/Time interval patterns for formatting date/time interval
dtptngen.h	C++ API: Date/Time Pattern Generator
dtrule.h	C++ API: Rule for specifying date and time in an year
fieldpos.h	C++ API: FieldPosition identifies the fields in a formatted output
fmtable.h	C++ API: Formattable is a thin wrapper for primitive types used for formatting and parsing
format.h	C++ API: Base class for all formats
formattednumber.h	C API: Formatted number result from various number formatting functions
formattedvalue.h	C++ API: Abstract operations for localized strings
fpositer.h	C++ API: FieldPosition Iterator
gender.h	C++ API: GenderInfo computes the gender of a list
gregocal.h	C++ API: Concrete class which provides the standard calendar
listformatter.h	C++ API: API for formatting a list
measfmt.h	C++ API: Compatibility APIs for measure formatting
measunit.h	C++ API: A unit for measuring a quantity
measure.h	C++ API: MeasureUnit object
messageformat2.h	C++ API: Formats messages using the draft MessageFormat 2.0
messageformat2_arguments.h	C++ API: Formats messages using the draft MessageFormat 2.0
messageformat2_data_model.h
messageformat2_data_model_names.h
messageformat2_formattable.h
messageformat2_function_registry.h
msgfmt.h	C++ API: Formats messages in a language-neutral way
nounit.h	C++ API: units for percent and permille
numberformatter.h	C++ API: All-in-one formatter for localized numbers, currencies, and units
numberrangeformatter.h	C++ API: Library for localized formatting of number, currency, and unit ranges
numfmt.h	C++ API: Compatibility APIs for number formatting
numsys.h	C++ API: NumberingSystem object
plurfmt.h	C++ API: PluralFormat object
plurrule.h	C++ API: PluralRules object
rbnf.h	C++ API: Rule Based Number Format
rbtz.h	C++ API: Rule based customizable time zone
regex.h	C++ API: Regular Expressions
region.h	C++ API: Region classes (territory containment)
reldatefmt.h	C++ API: Formats relative dates such as "1 day ago" or "tomorrow"
scientificnumberformatter.h	C++ API: Formats in scientific notation
search.h	C++ API: SearchIterator object
selfmt.h	C++ API: SelectFormat object
simplenumberformatter.h	C++ API: Simple number formatting focused on low memory and code size
simpletz.h	C++ API: SimpleTimeZone is a concrete subclass of TimeZone
smpdtfmt.h	C++ API: Format and parse dates in a language-independent manner
sortkey.h	C++ API: Keys for comparing strings multiple times
stsearch.h	C++ API: Service for searching text based on RuleBasedCollator
tblcoll.h	C++ API: The RuleBasedCollator class implements the Collator abstract base class
timezone.h	C++ API: TimeZone object
tmunit.h	C++ API: time unit object
tmutamt.h	C++ API: time unit amount object
tmutfmt.h	C++ API: Format and parse duration in single time unit
translit.h	C++ API: Transforms text from one format to another
tzfmt.h	C++ API: TimeZoneFormat
tznames.h	C++ API: TimeZoneNames
tzrule.h	C++ API: Time zone rule classes
tztrans.h	C++ API: Time zone transition
ucal.h	C API: Calendar
ucol.h	C API: Collator
ucoleitr.h	C API: UCollationElements
ucsdet.h	C API: Charset Detection API
udat.h	C API: DateFormat
udateintervalformat.h	C API: Format a date interval
udatpg.h	C API: Wrapper for icu::DateTimePatternGenerator (unicode/dtptngen.h)
udisplayoptions.h	C API: Display options (enum types, values, helper functions)
ufieldpositer.h	C API: UFieldPositionIterator for use with format APIs
uformattable.h	C API: UFormattable is a thin wrapper for primitive types used for formatting and parsing
uformattednumber.h	C API: Formatted number result from various number formatting functions
uformattedvalue.h	C API: Abstract operations for localized strings
ugender.h	C API: The purpose of this API is to compute the gender of a list as a whole given the gender of each element
ulistformatter.h	C API: Format a list in a locale-appropriate way
ulocdata.h	C API: Provides access to locale data
umsg.h	C API: MessageFormat
unirepl.h	C++ API: UnicodeReplacer
unum.h	C API: Compatibility APIs for number formatting
unumberformatter.h	C API: Localized number formatting; not recommended for C++
unumberoptions.h	C API: Header-only input options for various number formatting APIs
unumberrangeformatter.h	C API: Localized number range formatting
unumsys.h	C API: UNumberingSystem, information about numbering systems
upluralrules.h	C API: Plural rules, select plural keywords for numeric values
uregex.h	C API: Regular Expressions
uregion.h	C API: URegion (territory containment and mapping)
ureldatefmt.h	C API: URelativeDateTimeFormatter, relative date formatting of unit + numeric offset
usearch.h	C API: StringSearch
usimplenumberformatter.h	C API: Simple number formatting focused on low memory and code size
uspoof.h	C API: Unicode Security and Spoofing Detection
utmscale.h	C API: Universal Time Scale
utrans.h	C API: Transliterator
vtzone.h	C++ API: RFC2445 VTIMEZONE support
io
unicode
ustdio.h	C API: Unicode stdio-like API
ustream.h	C++ API: Unicode iostream like API
layoutex
layout
ParagraphLayout.h	C++ API: Paragraph Layout
playout.h	C API for paragraph layout
plruns.h	C API for run arrays
RunArrays.h	C++ API: base class for building classes which represent data that is associated with runs of text