Phenomena on the syntax-semantics interface of natural languages have been observed to have links with programming language semantics, namely computa- tional effects and evaluation order. We explore this connection to be able to profit from recent development in the study of effects. We propose adopting algebraic effects and handlers as tools for facilitating a uniform and integrated treatment of different non-compositional phenomena on the syntax-semantics interface. In this paper, we give an exposition of the framework of algebraic effects and handlers with an eye towards its applicability in computational semantics. We then present some exemplary analyses in the framework: we study the interplay of anaphora and quantification by translating the continuation-based dynamic logic of de Groote into a more DRT-like theory and we propose a treatment of overt wh-movement which avoids higher-order types in the syntax.