This is the first text to be published on stochastic Finslerian geometry. The theory is rigorously presented and several applications in ecology, evolution and epidemiology are described. Amongst the various topics covered are the role of curvature in Finslerian diffusions, Nelson's stochastic mechanics, nonlinear (Finslerian) filtering and entropy production. Two appendices deal with, respectively, the stochastic Hodge theory of Finslerian harmonic forms, and the theory of 2-dimensional Finsler spaces. The latter plays an important role in the applications described in the text.