The present invention provides pharmaceutically active pyrrolo[2,3-d]pyrimidinyl and pyrrolo[2,3-d]pyridinyl acrylamides and analogues thereof, having the structure: or a pharmaceutically acceptable salt or solvate thereof, or an enantiomer or diastereomer thereof, as set forth in the Description. Such compounds are useful for inhibiting Janus Kinase (JAK). This invention also is directed to compositions comprising methods for making such compounds, and methods for treating and preventing conditions mediated by JAK.