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The problem of enumerating (i.e., generating) all maximal cliques in a graph has received extensive treatment, due to the plethora of applications in various areas such as data mining, bioinformatics, network analysis and community detection. However, requiring the enumerated subgraphs to be full cliques is too restrictive in common real-life scenarios where "almost cliques" are equally useful. Hence, the notion of a k-plex, a clique relaxation that allows every node to be "missing" k neighbors, has been introduced. But this seemingly minor relaxation casts existing algorithms for clique enumeration inapplicable, for inherent reasons. This paper presents the first provably efficient algorithms, both for enumerating the maximal k-plexes and for enumerating the maximal connected k-plexes. Our algorithms run in polynomial delay for a constant k and incremental FPT delay when k is a parameter. The importance of such …