Non-relativistic Newton-Cartan gravity theories have attracted some attention recently in the context of condensed matter physics and Lifshitz holography. In this talk I will outline these motivations for studying Newton-Cartan gravity and suitable extensions thereof. I will then give an introduction to Newton-Cartan gravity and show how it can be obtained by considering a gauging of the Bargmann algebra. I will show how this procedure can be applied to the Schroedinger algebra to obtain torsional Newton-Cartan gravity, and to the super-Bargmann algebra to obtain three-dimensional Newton-Cartan supergravity. Finally, I will outline a procedure that allows one to obtain Newton-Cartan (super)gravity from relativistic (super)gravity.