It has been customary practice in galactic dynamics to implicitly assume that the corresponding N-body problem is (near) integrable. After a review of some relevant ideas from non-linear dynamics, I discuss the evidence suggesting that the above assumption is not generally satisfied and the consequences of such a situation. Next, I discuss the characterization of such "chaotic" behaviour. A geometric method-which I argue is best suited for measuring the instability properties of N-body systems-is tested on systems of 231 particles integrated with high precision and displaying "obvious" instabilities like violent relaxation and collective processes. The predicted instability time-scales show good agreement with those inferred from the spatial evolution. As a further test I study closed systems which relax towards definite equilibrium states. The times of relaxation towards such states are then compared to the exponential instability time-scales in an attempt to identify the physical interpretation of the exponential instability that appears to be always present in N -body systems. As an application of the method, the variation of the exponential divergence time-scales in N -body Plummer models with particle number, rotation, softening, and central mass is studied. I also study the extent of chaotic behaviour in some non-axisymmetric but smooth potentials representing galaxies with triaxial halos. This is done with the aid of Liapunov exponents, Poincare maps, and stability analysis of resonant orbits. It is found that a significant amount of chaos is usually present and increases dramatically with the addition of rotating bar perturbations, or of central masses. The degree of instability may also depend on the presence of external noise. It is also shown that dissipative perturbations have the important effect of producing an inflow of matter to the central areas. The consequences of the above processes are then discussed and it is suggested that they may explain some aspects of the observed relative bulge-disk-halo contributions to galaxy rotation curves.