Given an A-module M, and a dimension vector d, one can define a quiver Grassmannian, a projective algebraic variety parametrising the d-dimensional A-submodules of M. A famous result in geometric representation theory, obtained by several different authors, states that every projective variety X (over an algebraically closed field of characteristic zero) is isomorphic to such a quiver Grassmannian. In this talk I will explain how, at least in certain cases, one can use this algebraic description to construct a desingularisation of X. The construction is representation-theoretic, involving a tilt of an endomorphism algebra in mod(A), and the desingularising variety is again described in terms of quiver Grassmannians. This talk is based on joint work with Julia Sauter, in which we extend methodology of Crawley-Boevey and Sauter, and of Cerulli Irelli, Feigin and Reineke.