In approximation theory there are many classical and basic results on how well functions in certain function spaces (say smooth functions) can be approximated by elements from finite-dimensional subspaces (say e.g. splines; such an approximation is the simplest form of a numerical discretization). However, if the function to be approximated maps into a nonlinear manifold, what would be the corresponding discrete approximations, how does one compute them, and how does one show their approximation properties? There are several possibilities, and I will discuss one or two exemplary ones (most likely cubic spline curves).