Frequent connected subgraph mining (FCSM) has been an active area of research over the last twenty years. This review shall focus on the practical and theoretical issues arising in the *transactional setting*, where we are given a finite list of small to medium sized graphs and must find all graphs that are *subgraph isomorphic* to some user-defined number of graphs in the list. In particular, we present the generic approach to FCSM and investigate sufficient conditions for its computational tractability and intractability. Interestingly, it turns out that these are dependent on the complexity of the HamiltonianPath problem. This implies that FCSM is computationally tractable only for very restricted transaction graph classes. We subsequently review existing exact FCSM algorithms with a focus on applicability to arbitrary graph databases and present recent approximative FCSM algorithms that remain computationally tractable for all transactional databases.