This dissertation is a contribution to the differential-geometric treatment of classical field theories. In particular, I study both discrete and continuous aspects of classical field theories, in particular those with nonholonomic constraints. After some introductory chapters dealing with the geometric structures inherent in field theories and the discretization of field theories, the first part of the thesis is concerned with discrete field theories taking values in Lie groupoids. It is shown that many previously known discrete field theories are particular instances of Lie groupoid field theories, and the geometry of Lie groupoids is used to construct a unifying framework for this class. In two further chapters, the effect of symmetry upon this setup is described, with particular attention to the case of Euler-Poincaré reduction, which can be rephrased using concepts of discrete differential geometry. In the second part of the thesis, nonholonomic constraints for field theories are described. A number of differential-geometric results that characterize the nature of nonholonomic constraints are derived: in particular, a version of the De Donder-Weyl equation suitable for constrained field theories is discussed and a so-called momentum lemma is derived (describing the influence of symmetry upon the nonholonomic framework). In the last chapter, a physical example of a nonholonomic field theory is given, based on the theory of Cosserat media. This example is treated using the theory of the preceding chapters. Furthermore, a geometric numerical integration scheme is derived and used to give a quantitative insight into the dynamics
