A number of compilers from various vendors or open source communities implement the OpenMP API. If we are missing any please Contact Us with your suggestions.

Vendor/Source Compiler Information
Open Source GCC Free and open source – Linux, Solaris, AIX, MacOSX, Windows, FreeBSD, NetBSD, OpenBSD, DragonFly BSD, HPUX, RTEMS
From GCC 4.2.0, OpenMP 2.5 is fully supported.
From GCC 4.4.0, OpenMP 3.0 is fully supported.
From GCC 4.7.0, OpenMP 3.1 is fully supported.
In GCC 4.9.0, OpenMP 4.0 is supported for C and C++, but not Fortran.
From GCC 4.9.1, OpenMP 4.0 is fully supported.
From GCC 6.1, OpenMP 4.5 is fully supported for C and C++.
Compile with -fopenmp to enable OpenMP.
Online documentation:  https://gcc.gnu.org/onlinedocs/libgomp/
OpenMP support history: https://gcc.gnu.org/projects/gomp/
IBM XL C/C++ / Fortran XL C/C++ for AIX V13.1.3 and XL Fortran for AIX V15.1.3 support OpenMP 3.1 and selected features in OpenMP 4.0.

XL C/C++ for Linux V13.1.4 and XL Fortran for Linux V15.1.4 support OpenMP 3.1 and selected features in OpenMP 4.0.

XL C/C++ for Linux V13.1.5 on little endian distributions and XL Fortran for Linux V15.1.15 on little endian distributions (available in Dec 2016) support OpenMP 3.1 and features in OpenMP 4.5 (include device constructs for offloading to NVIDIA GPU).

XL C/C++
XL Fortran

XL compilers on POWER community:   C/C++ – Fortran

Oracle C/C++ / Fortran Oracle Developer Studio 12.5 compilers (C, C++, and Fortran) support OpenMP 4.0 features. Compile with -xopenmp.

Platforms: x86/Linux, x86/Solaris, SPARC/Solaris

Tools: Dbx debugger, Thread Analyzer for detecting data races and deadlocks, Code Analyzer for static code analysis, Performance Analyzer for performance profiling

Online Documentation
Free Download

Intel C/C++ / Fortran Windows, Linux, and MacOSX.
OpenMP 3.1 C/C++/Fortran fully supported in version 12.0, 13.0, 14.0 compilers
OpenMP 4.0 C/C++/Fortran supported in version 15.0 and 16.0 compilers
OpenMP 4.5 C/C++/Fortran supported in version 17.0 compilers
Compile with -Qopenmp on Windows, or just -openmp or -qopenmp on Linux or Mac OSX
More detailed information
PGI C/C++ / Fortran Support for OpenMP 3.1 in all languages on x86 systems running Linux and OS X, OpenPOWER systems running Linux, and in Fortran and C on Windows/x86.  Compile with -mp to enable OpenMP on all platforms.

More Information on PGI Compilers

Absoft Pro Fortran Fortran Versions 11.1 and later of the Absoft Fortran 95 compiler for Linux, Windows and Mac OS X include integrated OpenMP 3.0 support. Version 17.0 supports OpenMP 3.1.  Compile with -openmp.
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Lahey/Fujitsu Fortran 95 C/C++ / Fortran The compilers in the software package of ‘Technical Computing Suite for the PRIMEHPC FX100′ support OpenMP 3.1.

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PathScale C/C++ / Fortran Linux 32/64 bit.EKOPath 6: http://www.pathscale.com/EKOPath – CPU only. Fully supports OpenMP 2.5. Supports almost all OpenMP 3.x and 4.0, no OpenMP4 offload directives support.
ENZO2016: http://www.pathscale.com/ENZO – CPU+GPU – Fully supports OpenMP 2.5. Supports almost all OpenMP 3.x and 4.0, includes OpenMP4 offloading
Cray Cray C/C++ and Fortran Cray Compiling Environment (CCE) 8.5 (June 2016) supports OpenMP 4.0, with OpenMP 4.5 support for device constructs. OpenMP is on by default.
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NAG nagfor NAG Fortran Compiler 6.1 supports OpenMP 3.1 on x86 and x64, for Linux, Mac and Windows. Compile with –openmp.
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OpenUH Research Compiler C/C++/Fortran The OpenUH 3.x compiler has a full open-source implementation of OpenMP 2.5 and near-complete support for OpenMP 3.0 (including explicit task constructs) on Linux 32-bit or 64-bit platforms. For more information or to download: http://web.cs.uh.edu/~openuh/index.shtml
LLVM clang Clang is an open-source (permissively licensed) C / C++ compiler that is available to download at http://llvm.org/releases/download.html. The Clang 3.9 release supports all non-offloading features of OpenMP 4.5. Offloading support is under development and expected to become available in a future version.

Compile and link your code with -fopenmp

LLNL Rose Research Compiler C/C++/Fortran ROSE is a source-to-source research compiler supporting OpenMP 3.0 and some OpenMP 4.0 accelerator features targeting NVIDIA GPUs.

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Appentra Solutions parallware compiler Parallware Suite C/C++ Parallware Suite provides the tool Parallware Trainer, an interactive, real-time desktop tool that facilitates teaching, learning, and the usage of parallel programming using directives of OpenMP 4.0.

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Texas Instruments C
OpenMP 3.0 is supported on TI’s Keystone I family of Multicore C66x Digital Signal Processor (DSP) SoCs using the Multicore Software Development Kit MCSDK-C66 2.1.2.6.
OpenMP 3.0 with the device constructs from OpenMP 4.0 is supported on TI’s Keystone II family of C66x+Cortex-A15 SoCs using MCSDK-K2 3.1.4.7
Barcelona Supercomputing Center
Mercurium C/C++/Fortran Mercurium is a source-to-source research compiler that is available to download at https://github.com/bsc-pm/mcxx

OpenMP 3.1 is almost fully supported for C,  C++, Fortran. Apart from that,  almost all tasking features introduced in newer versions of OpenMP are also supported. More information