Random polytopes have a long history, going back to Sylvester's famous four-point problem of the 19th century.  Since then their study has become a mainstream topic in convex and stochastic geometry, with close connection to polytopal approximation problems, among other things.  In this talk we will consider random polytopes in constant curvature geometries, and show that their volume satisfies a central limit theorem.  The proof uses Stein's method for normal approximation, and extends to general projective Finsler metrics.