Deterministic Suffix-reading Automata (DSA) are a model of automata over finite words introduced recently. DSAs are able to represent regular languages more succinctly than conventional DFAs.
Transitions in a DSA are labeled with words. From a state, a DSA triggers an outgoing transition on seeing a word ending with the transition's label. Therefore, rather than moving along an input word letter by letter, a DSA can jump along blocks of letters, with each block ending in a suitable suffix. This enables DSAs to represent regular languages more concisely.
This is a very recent model with plenty of open questions. Here is one concrete question: given a DSA, is it minimal?
For DFAs, this question can be answered by identifying a distinguishing string for every pair of states, thanks to the Myhill-Nerode theorem. No such characterization is known in the DSA setting.