The question addressed is the determination of a glacier’s subglacial topography, given surface topography and mass-balance data. The input data can be obtained relatively easily for a large number of glaciers. Several methods essentially based on the shallow ice approximation are proposed, some of which are extended to Stokes ice flows. Two gradient-free, iterative methods are first introduced, namely the quasi-stationary inverse method, that relies on the apparent surface mass-balance description of glacier dynamics, and the transient inverse method, consisting in the iterative update of the bedrock topography proportionally to the surface topography misfit at the end of the glacier’s considered evolution. Then, an optimal control algorithm is suggested that calculates the bedrock topography and some model parameters from surface observations through the minimization of a regularized misfit functional by means of a Lagrangian method. Numerical validations, along with sensitivity analyses and applications to real-world data are presented for each method.