  Applied to field theory, the familiar symplectic technique leads to
instantaneous Hamiltonian formalism on an infinite-dimensional phase space. A
true Hamiltonian partner of first order Lagrangian theory on fibre bundles
$Y\to X$ is covariant Hamiltonian formalism in different variants, where
momenta correspond to derivatives of fields relative to all coordinates on $X$.
We follow polysymplectic (PS) Hamiltonian formalism on a Legendre bundle over
$Y$ provided with a polysymplectic $TX$-valued form. If $X=\mathbb R$, this is
a case of time-dependent non-relativistic mechanics. PS Hamiltonian formalism
is equivalent to the Lagrangian one if Lagrangians are hyperregular. A
non-regular Lagrangian however leads to constraints and requires a set of
associated Hamiltonians. We state comprehensive relations between Lagrangian
and PS Hamiltonian theories in a case of semiregular and almost regular
Lagrangians. Quadratic Lagrangian and PS Hamiltonian systems, e.g. Yang - Mills
gauge theory are studied in detail. Quantum PS Hamiltonian field theory can be
developed in the frameworks both of familiar functional integral quantization
and quantization of the PS bracket.
