This work reviews the classical Darboux theorem for symplectic,
presymplectic, and cosymplectic manifolds (which are used to describe regular
and singular mechanical systems), and certain cases of multisymplectic
manifolds, and extends it in new ways to k-symplectic and k-cosymplectic
manifolds (all these structures appear in the geometric formulation of
first-order classical field theories). Moreover, we discuss the existence of
Darboux theorems for classes of precosymplectic, k-presymplectic,
k-precosymplectic, and premultisymplectic manifolds, which are the geometrical
structures underlying some kinds of singular field theories. Approaches to
Darboux theorems based on flat connections associated with geometric structures
are given, while new results on polarisations for (k-)(pre)(co)symplectic
structures arise.Comment: improved and extended proofs. 33 p