The present work investigates a number of problems related to the modelling and approximation of nonlinear systems, using geometry as the primary lens through which ideas are explored. The first part of the work focuses on the fundamental problems of system identification and model reduction for nonlinear systems. Three different approaches to the identification of nonlinear systems are developed using nonlinear realization theory, ideas from subspace identification and functional equations. The model reduction problem at isolated singularities is then posed and solved using the concept of moment matching. Motivated by these results, the second part of the work develops several notions and tools for modelling nonlinear systems. First, a nonlinear enhancement of the notions of eigenvalue and of pole is introduced and studied exploiting the differential geometric approach to nonlinear systems. The persistence of excitation of signals generated by autonomous systems is then characterized in geometric terms. Finally, connections between moments of systems and moments of random variables are established. The theory is illustrated by means of several examples and the applicability of the resulting algorithms is verified by numerical simulations.