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In this seminar I will give a non-specialist introduction into the non-relativistic version of Riemannian geometry, which is called NewtonCartan geometry. The corresponding non-relativistic gravity theory, called Newton-Cartan gravity, has received a renewed attention due to unexpected applications as a new tool to examine the nonperturbative properties of non-relativistic quantum field theories. I will explain the basic construction of Newton-Cartan gravity plus some of its extensions including one with non-relativistic higher spins in three spacetime dimensions.

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