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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Applied to classical field theory, the familiar symplectic Hamiltonian formalism takes the form of instantaneous Hamiltonian formalism on an infinite-dimensional phase space, where canonical coordinates are field functions at some instant of time. This Hamiltonian formalism is applied to quantization of fields, e.g., in quantum gauge theory. The true Hamiltonian counterpart of classical first order Lagrangian field theory is covariant Hamiltonian formalism where canonical momenta p^mu_i correspond to derivatives of fields with respect to all world coordinates x . Covariant Hamilton equations are equivalent to the Euler-Lagrange equations in the case of hyperregular Lagrangians. Covariant Hamiltonian field theory is developed in the Hamilton-De Donder, polysymplectic, multisymplectic and k-symplectic variants. A phase space of covariant Hamiltonian field theory is a finite-dimensional polysymplectic or multisymplectic manifold.