Approval-based scoring rules require participants to submit ballots that correspond to the candidates they approve of, and each candidate then receives some score from each ballot it appears on. One particularly natural class of approval-based scoring rules is the class of size-approval rules. For rules in this class, each ballot is weighted based on the number of candidates it contains. We provide a deep axiomatic analysis of the class of size-approval rules and related classes of approval-based scoring rules. We organise existing work by shedding light on the exact role played by core axioms involved in previous characterisations of size-approval rules, while also providing multiple new characterisations of the same class and its closest relatives. We furthermore provide an original characterisation of even-and-equal cumulative voting, the most paradigmatic representative of the size-approval voting rules, where each ballot's weight is inversely proportional to its size.