  Genuine entanglement is the strongest form of multipartite entanglement.
Genuinely entangled pure states contain entanglement in every bipartition and
as such can be regarded as a valuable resource in the protocols of quantum
information processing. A recent direction of research is the construction of
genuinely entangled subspaces -- the class of subspaces consisting entirely of
genuinely multipartite entangled pure states. In this paper we present several
methods of construction of such subspaces including those of maximal possible
dimension. The approach is based on the correspondence between bipartite
entangled subspaces and quantum channels of a certain type. The examples
include maximal subspaces for systems of three qubits, four qubits, three
qutrits. We also provide lower bounds on two entanglement measures for mixed
states, the concurrence and the convex-roof extended negativity, which are
directly connected with the projection on genuinely entangled subspaces.
