Motifs are the fundamental components of complex systems. The topological structure of networks representing complex systems and the frequency and distribution of motifs in these networks are intertwined. The complexities associated with graph and subgraph isomorphism problems, as the core of frequent subgraph mining, have direct impacts on the performance of motif discovery algorithms. To cope with these complexities, researchers have adopted different strategies for candidate generation and enumeration, and frequency computation. In the past few years, there has been an increasing interest in the analysis and mining of temporal networks. These networks, in contrast to their static counterparts, change over time in the form of insertion, deletion, or substitution of edges or vertices or their attributes. In this paper, we provide a survey of motif discovery algorithms proposed in the literature for mining static and temporal networks and review the corresponding algorithms based on their adopted strategies for candidate generation and frequency computation. As we witness the generation of a large amount of network data in social media platforms, bioinformatics applications, and communication and transportation networks and the advance in distributed computing and big data technology, we also conduct a survey on the algorithms proposed to resolve the CPU-bound and I/O bound problems in mining static and temporal networks.