  We introduce a natural notion of quaternionic map between almost quaternionic
manifolds and we prove the following, for maps of rank at least one: 1) A map
between quaternionic manifolds endowed with the integrable almost twistorial
structures is twistorial if and only if it is quaternionic. 2) A map between
quaternionic manifolds endowed with the nonintegrable almost twistorial
structures is twistorial if and only if it is quaternionic and
totally-geodesic. As an application, we describe the quaternionic maps between
open sets of quaternionic projective spaces.
