A connection is a geometric object that allows to parallel transport vectors along a curve in a domain. A natural question that often arises is whether one can recover a connection inside a domain from the knowledge of the parallel transport along a set of special curves running between boundary points of the domain. In this talk I will discuss this geometric inverse problem in various settings including Riemannian manifolds with boundary and Minkowski space. This problem is related to other inverse problems and is tackled with a range of techniques that I will explore during the talk.