Crucial nodes in a network refer to those nodes that their existence is so important in preserving topological structure of the network and they independently determine the network structure. In this study I introduced and proposed several mathematical methods for identifying crucial nodes in networks. They fall into three categories, node perturbation, network analysis, and network dynamics. Node perturbation methods include adjacency matrix index, degree or flow change index, node perturbation index, etc. Network dynamics methods include network evolution modeling, etc. Network analysis methods include node degree, criticality index, branch flourishing index, node importance index, etc. Advantages and advantages of these methods were discussed. Finally, I suggested that some of these methods may also be used to identify crucial links(connections) in networks. In this case, the change of a link refers to presence/absence of a link, or change of flow in the link, etc.