In this expository paper, the author addresses the question of constructing parallelisms with some degree of symmetry, namely he considers resolvable block designs admitting an automorphism group acting doubly transitively on the resolution classes. This situation is examined in different contexts, in particular he surveys results for finite affine planes and spaces, for finite circle planes (Möbius, Laguerre and Minkowski finite planes respectively) and for finite complete graphs with resolution classes given by 1-factors. A complete and useful bibliography is contained in the paper.