Part I of this thesis defines a Laplacian A for a Finsler space; we obtain A by requiring that ( A f) ( x) for a function f measures the infinitesimal average of f around x. This A is a linear, elliptic, 2nd-order differential operator. Furthermore, Af can be written in a divergence form, like the Riemannian Laplacian, but with respect to a canonical osculating Riemannian metric and Busemann's intrinsic volume form. We interpret divergence form as the result of minimizing a certain energy functional on Finsler space, and further use this approach to define harmonic forms, and harmonic mappings between Finsler manifolds. As a byprod-uct of the Laplacian, in Part I1 we derive a simple volume-form inequality which characterizes Riemannian manifolds, and define a scalar invariant V ( x) for Finsler spaces. We show that, on a Berwald space, the met-ric's first derivatives vanish in normal co-ordinates, and use that result t
