Sesqui-pushout (SqPO) rewriting - "sesqui" means "one and a half" in Latin - is a new algebraic approach to abstract rewriting in any category. SqPO rewriting is a deterministic and conservative extension of double-pushout (DPO) rewriting, which allows to model "deletion in unknown context", a typical feature of single-pushout (SPO) rewriting, as well as cloning. After illustrating the expressiveness of the proposed approach through a case study modelling an access control system, we discuss sufficient conditions for the existence of final pullback complements and we analyze the relationship between SqPO and the classical DPO and SPO approaches.