A number of recent significant developments in the theory of differential equations are presented in an elementary fashion, many of which are scattered throughout the literature and have not previously appeared in book form, the common denominator being the theory of planar vector fields (real or complex). A second common feature is the study of bifurcations of dynamical systems. Moreover, the book links fields that have developed independently and signposts problems that are likely to become significant in the future. The following subjects are covered: new tools for local and global properties of systems and families of systems, nonlocal bifurcations, finiteness properties of Pfaffian functions and of differential equations, geometric interpretation of the Stokes phenomena, analytic theory of ordinary differential equations and complex foliations, applications to Hilbert's 16th problem.