  In contrast with QFT, classical field theory can be formulated in a strict
mathematical way if one defines even classical fields as sections of smooth
fiber bundles. Formalism of jet manifolds provides the conventional language of
dynamic systems (nonlinear differential equations and operators) on fiber
bundles. Lagrangian theory on fiber bundles is algebraically formulated in
terms of the variational bicomplex of exterior forms on jet manifolds where the
Euler--Lagrange operator is present as a coboundary operator. This formulation
is generalized to Lagrangian theory of even and odd fields on graded manifolds.
Cohomology of the variational bicomplex provides a solution of the global
inverse problem of the calculus of variations, states the first variational
formula and Noether's first theorem in a very general setting of
supersymmetries depending on higher-order derivatives of fields. A theorem on
the Koszul--Tate complex of reducible Noether identities and Noether's inverse
second theorem extend an original field theory to prequantum field-antifield
BRST theory. Particular field models, jet techniques and some quantum outcomes
are discussed.
