Several algorithms have been recently designed to identify motifs in biological networks, particularly in protein-protein interaction networks. Motifs correspond to repeated modules in the network that may be of biological interest. The approaches proposed in the literature often differ in the definition of a motif, the way the occurrences of a motif are counted and the way their statistical significance is assessed. This has strong implications on the computational complexity of the discovery process and on the type of results that can be expected. This review presents in a systematic way the different computational settings outlining their main features and limitations.