This monograph discusses a topic that has developed from original work of L.S. Pontryagin. In 1961, at the first IFAC Congress in Moscow, he presented his maximum principle (PMP). At the same Congress, A.T. Fuller exhibited an example of optimal control problem in which the optimal control is not piecewise continuous but is measurable only: that is, there exists an infinite number of switches on a finite time interval (Fuller"s phenomenon). This fact obstructs the direct use of PMP and the problem was solved only due to its specific symmetry. Fuller"s example was interpreted as a curious pathology and soon forgotten. Most recently, it was proved that Fuller"s phenomenon is that of general position. In spite of its seemingly artificial, purely theoretical nature, this phenomenon is inherent in a great diversity of interesting nonlinear optimal control problems of applied content. This book presents a systematic treatment of fields of optimal trajectories with an infinite number of switches on a finite time interval. A method of resolution of singularities of the first return Poincare mapping for discontinuous Hamiltonian systems, developed in the book, allows the solution of problems with Fuller"s phenomenon which defy the usual methods. Lagrangian manifolds and optimal syntheses, including extremals with an infinite number of switches, are designed. While the authors break new theoretical ground, the book is written to be accessible to a wider audience, including advanced students and engineers. Special attention is focused on the heuristic and algorithmic items. In addition, recipes for solving concrete problems are given.