The modern subject of Dynamical Systems concerns the study of the orbit structure of Semi-Group actions on various spaces. Until the latter part of the 19th century, this study was primarily restricted to the kinematics of rigid motions and the dynamics of classical mechanics. Of principal concern were problems in Celestial Mechanics. Poincare discovered many new phenomena regarding periodic motions in the Restricted Three Body Problem--in particular homoclinic orbits. These are orbits which are asymptotic to periodic orbits in forward and backward times. They turn out to be related to newer concepts such as hyperbolicity, topological entropy, and symbolic dynamics, which, even recently, give us substantial understanding of two dimensional diffeomorphisms and three dimensional flows. We describe some historical developments leading to these advances.