AbstractThis paper gives an overview of recent results concerning the modular derivation of (i) modal specification logics, (ii) notions of simulation together with logical characterisations, and (iii) sound and complete axiomatisations, for systems modelled as coalgebras of functors on Set. Our approach applies directly to an inductively-defined class of coalgebraic types, which subsumes several types of discrete state-based systems, including (probabilistic) transition systems, probabilistic automata and spatial transition systems.