We have introduced a very general scheme for defining crossover operators for graphs that start from cuts in the parental graphs and subsequently use a generalized join operation. This construction is very flexible and makes it possible to restrict the offspring to conserving certain properties of the parents. This makes cut-and-join crossover a generally applicable class of operators to implement recombination in genetic algorithms that operate on graphs. For applications to chemical graphs, degree-preservation and the preservation of graph-theoretical planarity are of particular relevance.