  In the context of irreducible holomorphic symplectic manifolds, we say that
(anti)holomorphic (anti)symplectic involutions are brane involutions since
their fixed point locus is a brane in the physicists' language, i.e. a
submanifold which is either complex or lagrangian submanifold with respect to
each of the three K\"ahler structures of the associated hyperk\"ahler
structure.
  Starting from a brane involution on a K3 or abelian surface, one can
construct a natural brane involution on its moduli space of sheaves. We study
these natural involutions and their relation with the Fourier--Mukai transform.
Later, we recall the lattice-theoretical approach to Mirror Symmetry. We
provide two ways of obtaining a brane involution on the mirror and we study the
behaviour of the brane involutions under both mirror transformations, giving
examples in the case of a K3 surface and $K3^{[2]}$-type manifolds.
