  Realizations of $\kappa$-Minkowski space linear in momenta are studied for
time-, space- and light-like deformations. We construct and classify all such
linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators.
There are three one-parameter families of linear realizations for time-like and
space-like deformations, while for light-like deformations, there are only four
linear realizations. The relation between deformed Heisenberg algebra, star
product, coproduct of momenta and twist operator is presented. It is proved
that for each linear realization there exists Drinfeld twist satisfying
normalization and cocycle conditions. $\kappa$-deformed
$\mathfrak{igl}(n)$-Hopf algebras are presented for all cases. The
$\kappa$-Poincar\'e-Weyl and $\kappa$-Poincar\'e-Hopf algebras are discussed.
Left-right dual $\kappa$-Minkowski algebra is constructed from the transposed
twists. The corresponding realizations are nonlinear. All known Drinfeld twists
related to $\kappa$-Minkowski space are obtained from our construction.
Finally, some physical applications are discussed.
