Abstract. The algebraic approach to graph transformation has been based on the understanding of graphs as many-sorted unary total algebras, and it has even been generalized to such algebras. However, it seems natural to interpret graphs as two-sorted partial algebras with countably many possible source and target operations. In this paper, graph transformation is studied from the point of view of partial algebras. The richness of the theory of partial algebras entails the existence of different approaches to double-pushout and simple-pushout transformation, using different notions of graph morphism, two of which correspond to the usual double-pushout and single-pushout approaches to graph transformation, while the other approaches are novel.