Combinatorial reconfiguration problems that have been traditionally discussed in discrete mathematics will be presented with an emphasis on the speaker's work and open problems, as well as computational aspects that have recently been introduced. Key concepts include the Hasse diagrams of partially ordered sets, the 1-skeleta of convex polytopes, and the Cayley graphs of finite groups. Those concepts will be introduced with an example of sorting, a fundamental problem in computer science. Then, connections of Catalan structures, linear programming and permutation groups with with combinatorial reconfiguration will be discussed.