This contribution is organized as follows: After introducing some basic graphtheoretic notation and discussing properties of chemical graphs, we give a formal account of the class of cut-and-join crossover operations. In particular, we show under which conditions degree-preserving and bond-order-preserving joins exist. We proceed by showing that under these conditions it is also always possible to construct crossover offspring that preserve graph-theoretical planarity of the structural formula. We then consider the problem of computing all recombinant offspring for a pair of parents for degree-or bond-order preserving operations. Continuing the analysis of the operators, we proceed to show that already in the degree-preserving case, cut-and-join crossover is sufficient to generate any molecular graph in a small population starting from a small molecular base set. In the empirical part of this work, we demonstrate that cut-and-join crossover produced highly diverse offspring that are nevertheless chemically plausible, synthesizable, and even drug-like with a sufficiently high probability that unrealistic proposal can be easily removed by well-established filtering steps. As a showcase application for cut-and-join crossover we describe its integration with REvoLd to search for ligands binding to four different target proteins. Finally, we close with a discussion of the main results and some open questions.