In this talk we give an overview of the geometry of moduli spaces of shtukas for the general linear groups GL(n). These moduli spaces have been instrumental in the work of Drinfeld and Lafforgue on the Langlands correspondence for GL(n) over function fields. We then explore the connection between the geometry of these moduli spaces and the wonderful compactification of De Concini-Procesi. This talk is based on joint work with Y. Varshavsky.