The following language bindings are provided and supported by The HDF Group.

CHDF5 is written in C. Building with configure or CMake will automatically include the C APIs.
FortranIncluded in the HDF5 source. Build by specifying --enable-fortran (configure) or -DHDF5_BUILD_FORTRAN:BOOL=ON (CMake)
C++Included in the HDF5 source code. Build by specifying --enable-cxx (configure) or -DHDF5_BUILD_CPP_LIB:BOOL=ON (CMake)
JavaIncluded in the HDF5 source code. Build by specifying--enable-java (configure) or -DHDF5_BUILD_JAVA:BOOL=ON (CMake)

See the Visual Studio Solution on Github that includes the declarations and unit tests. Also see the NuGet package that includes "Any CPU" target assembly plus the unmanaged dependencies


Users have also created their own bindings, some of which are included below.

AdaAda, Adapp: The HDF5 Library for HDF5-1.8.9 and HDF5-1.8.14 from IDRIS
C++H5CPP - HDF5 C++ templates for HDF5
C++h5cpp - HDF5 C++ wrapper for the HDF5 C library 
Common Lispcl-ana - Free (GPL) Common Lisp data analysis library with emphasis on modularity and conceptual clarity
DelphiDelphi bindings - available from the HDF Forum
MathematicaHDF5Mathematica - HDF5 interface for Mathematica
Javajhdf5 - Java binding for HDF5-1.8, focusing on ease-of-use and maintained by ETH SIS
JuliaJuliaIO/HDF5 - HDF5 interface for the Julia language
Pythonh5py - Pythonic interface to HDF5
Rrhdf5 - R interface for HDF5