Firstly, we introduce the notion of overlap functions on complete lattices and give two construction methods of them. Secondly, we show some basic properties of overlap functions on complete lattices. In particular, we extend the notions of migrativity and homogeneity of overlap functions on unit closed interval to the so-called migrativity and homogeneity of overlap functions on complete lattices, respectively. Finally, we give an analogous discussion for grouping functions on complete lattices.