The paper reviews recent results which aim at generalizing finite automata theory from words and trees to labelled partial orders (presented as labelled directed acyclic graphs), with an emphasis on logical aspects. As an important type of labelled partial order we consider pictures (two-dimensional words). Graph acceptors and their specialization for pictures, “tiling systems”, are presented, and their equivalence to existential monadic second-order logic is reviewed. Other restricted versions of graph acceptors are discussed, and an intuitive exposition of the recently established monadic quantifier alternation hierarchy over graphs is given.