In contrast with quantum theory, classical field theory can be formulated in a strict mathematical way by treating classical fields and dynamic variables as sections of smooth fiber bundles. This book aims to compile the relevant material on fiber bundles, jet manifolds, connections, graded manifolds and Lagrangian theory. The book is based on the graduate and post graduate courses of lectures given at the Department of Theoretical Physics of Moscow State University (Russia). It addresses to a wide audience of mathematicians, mathematical physicists and theoreticians.