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The GPL or LGPL licenses allows you those freedoms, so it is free software.
Keep in mind that Cygwin can only do as much as the underlying OS supports.
Because of this, Cygwin will behave differently, and exhibit different limitations, on the various versions of Windows.
The Cygwin distribution contains thousands of packages from the Open Source world including most GNU tools, many BSD tools, an X server and a full set of X applications.
If you're a developer you will find tools, headers and libraries allowing to write Windows console or GUI applications that make use of significant parts of the POSIX API.
State January 2016 this includes Windows Vista, Windows Server 2008 and all later versions of Windows up to Windows 10 and Windows Server 2016.