In this talk I will give two examples of inverse problems in glaciology. The first one states as follows: how to infer a climatic scenario (i.e. how to reconstruct past polar temperature) from ice volume records? Past works on the subject used fairly simple methods, such as nudging. The idea of this work is to explore the ability of the adjoint method to solve the inverse problem of reconstructing past temperature given all available observations. We start here with a simplified ice-sheet model and perform twin experiments. The second inverse problem is based on the following motivation: to compute the ice discharge of Antarctica and Greenland in the near future, thanks to simulations of polar ice-sheet models, in order to assess their impacts on sea-level change. As the ice discharge is closely linked to ice velocities, which are in turn highly sensitive to basal friction parameter, we applied two existing approximate inverse methods to estimate the (usually poorly known, because unobserved) basal drag parameter over Greenland.