  Classification of differential forms on $\kappa$-Minkowski space,
particularly, the classification of all bicovariant differential calculi of
classical dimension is presented. By imposing super-Jacobi identities we derive
all possible differential algebras compatible with the $\kappa$-Minkowski
algebra for time-like, space-like and light-like deformations. Embedding into
the super-Heisenberg algebra is constructed using non-commutative (NC)
coordinates and one-forms. Particularly, a class of differential calculi with
an undeformed exterior derivative and one-forms is considered. Corresponding NC
differential calculi are elaborated. Related class of new Drinfeld twists is
proposed. It contains twist leading to $\kappa$-Poincar\'e Hopf algebra for
light-like deformation. Corresponding super-algebra and deformed super-Hopf
algebras, as well as the symmetries of differential algebras are presented and
elaborated. Using the NC differential calculus, we analyze NC field theory,
modified dispersion relations, and discuss further physical applications.
