This work incorporates three modern aspects of mathematical physics: the jet methods in differntial geometry, the Lagrangian formalism on jet manifolds and the multimomentum approach to the Hamiltonian formalism. Several contemporary field models are investigated in detail. This is not a book on differential geometry, although modern concepts of differential geometry are used throughout the book. Quadratic Lagrangians and Hamiltonians are studied on the general level including the treatment of Hamiltonian formalism on composite fibre manifolds. The book presents new geometric methods and results in field theory.