This book provides an in-depth coverage of modern research on dynamical systems. The first part discusses stellar dynamics, integrable systems, the transition to chaos and instabilities in stellar dynamics as well as the dynamics of spiral galaxies. Models are given and compared with observations. The second part is devoted to the direct method of N-body simulations, to gas dynamics simulations and to galaxy formation. Special care is taken to give to a pedagogical presentation of the material which makes this a unique text well suited for graduate courses in astrophysics.