Solar magnetism is responsible for the several active phenomena that occur in the solar atmosphere. The consequences of these phenomena on the solar-terrestrial environment and on Space Weather are nowadays clearly recognized, even if not yet fully understood. In order to shed light on the mechanisms that are at the basis of the Space Weather, it is necessary to investigate the sequence of phenomena starting in the solar atmosphere and developing across the outer layers of the Sun and along the path from the Sun to the Earth. This goal can be reached by a combined multi-disciplinary, multi-instrument, multi-wavelength study of these phenomena, starting with the very first manifestation of solar active region formation and evolution, followed by explosive phenomena (i.e., flares, erupting prominences, coronal mass ejections), and ending with the interaction of plasma magnetized clouds expelled from the Sun with the interplanetary magnetic field and medium. This wide field of research constitutes one of the main aims of COST Action ES0803: Developing Space Weather products and services in Europe. In particular, one of the tasks of this COST Action was to investigate the Progress in Scientific Understanding of Space Weather. In this paper we review the state of the art of our comprehension of some phenomena that, in the scenario outlined above, might have a role on Space Weather, focusing on the researches, thematic reviews, and main results obtained during the COST Action ES0803.