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The x86-64 System V ABI (used on everything except Windows) used to live at http://x86-64.org/documentation/abi.pdf, but that site has now fallen off the internet.
Is there a new authoritative home for the document?
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The System V Application Binary Interface is a set of specifications that detail calling conventions, object file formats, executable file formats, dynamic linking semantics, and much more for systems that complies with the X/Open Common Application Environment Specification and the System V Interface Definition.
The AMD ABI allows all 64-bit instructions and 64-bit registers to be used to their full effect. Many of the new instructions are straightforward extensions of the existing i386 instruction set. There are now sixteen general purpose registers.
The System V x86-64 psABI document is maintained as LaTeX sources on GitLab. Similarly the i386 psABI is a separate GitLab repo. (Formerly on github). Those pages have info on where revisions are discussed.
(TODO: find a link for PDFs built from latest revisions. The links in the rest of this answer haven't been updated for a while.)
See also the x86 tag wiki for direct links to the latest versions.
As of now, the current version for x86-64 is 1.0 (January 2018). There's a more recent 2018-dec draft version which (ironically) removes the "Draft" from the "Draft 1.0" version number in the PDF itself.
The x32 ABI (32-bit pointers in long mode) is part of the x86-64 ABI doc. See Chapter 10: ILP32 Programming Model.
The current version for i386 is 1.1. (Note that some OSes use an older version of the i386 ABI which doesn't require 16-byte stack alignment, only 4. GCC ended up depending on -mpreferred-stack-boundary=4 16-byte alignment for its SSE code-gen (perhaps unintentionally), and eventually the ABI got updated for Linux to enshrine that as an official requirement. I attempted a summary in a comment on GCC bug #40838. This breaks backwards compat with some hand-written asm that calls other functions.)
Unofficially, sign-extending narrow args to 32-bit is required (for both i386 and amd64), because clang depends on it. Hopefully a future ABI revision will document that.
The Processor Supplement (psABI) docs are designed as a supplement to the less-frequently-updated System V gABI (generic), hosted on SCO's website.
Other links
Also https://refspecs.linuxfoundation.org/ hosts a copy of the gABI from 1997.
https://uclibc.org/specs.html has psABI links for various non-x86 ISAs. (Although for example the ARM one only seems to document the ELF file layout, not the calling convention or process startup state.) https://uclibc.org/docs/psABI-x86_64.pdf is an outdated copy of the x86-64 psABI (0.99.7 from 2014). The version on GitHub has clearer wording of a few things and bugfixes in some examples.
Related: What are the calling conventions for UNIX & Linux system calls (and user-space functions) on i386 and x86-64 describes the system-call calling convention for x86-64 SysV (as well as i386 Linux vs. FreeBSD).
It also summarizes the function calling conventions for integer args. System calls don't take FP or SSE/AVX vector args, or structs by value, so the function-calling convention is more complicated.
Agner Fog has a calling conventions guide (covering Windows vs. Sys V, and the various conventions for 32-bit, and tips/tricks for writing functions you can use on either platform). This is a separate PDF from his optimization and microarchitecture guides and instruction tables (which are essential reading if you care about performance.)
Wikipedia has an x86 calling conventions article which describes various conventions, but mostly not in enough detail to use them for anything other than simple integer args. (e.g. no description of struct-packing rules).
GCC and Clang (on all architectures) use the C++ ABI originally developed for Itanium. https://itanium-cxx-abi.github.io/cxx-abi/. This is relevant for example for what requirements a C++ struct/class need to be passed in registers (e.g. being an aggregate according to some definition).
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Linux standard base
The Linux Standard Base, which can be considered by some the authoritative spec for this matter, has section called 7.2. "Function Calling Sequence" points to the 2.1. "Normative References section" which contains the following links:
I would therefore recommend using those versions of the specifications as the canonical ones unless you have good reason to do otherwise.
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