Abstract Projection in the situation calculus refers to answering queries about the future evolutions of the modeled domain, while progression refers to updating the logical representation of the initial state so that it reflects the changes due to an executed action. In the general case projection is not decidable and progression may require second-order logic. In this paper we focus on a recent result about the decidability of projection and use it to drive results for the problem of progression. In particular we contribute with the following: (i) a major result showing that for a large class of intuitive action theories with bounded unknowns a first-order progression always exists and can be computed; (ii) a comprehensive classification of the known classes that can be progressed in first-order; (iii) a novel account of nondeterministic actions in the situation calculus.