We develop projective and conformal approaches to almost complex structures. In both cases, we replace the almost complex structure by a weighted analogue. In the conformal setting, using a certain canonical connection, we provide criteria characterising certain classes of the Gray-Hervella classification of almost Hermitian manifolds, notably the class of semi-Kaehler manifolds, which includes nearly and almost Kaehler manifolds as special cases. Similar results can be obtained in projective geometry. Time permitting, I will also cover applications to conformal Patterson-Walker geometries. This is joint work with Josef Šilhan.