Phylogenetic trees have been widely used in the study of evolutionary biology for representing the tree-like evolution of a collection of species. However, different data sets and different methods often lead to the construction of different phylogenetic trees for the same set of species. Therefore, comparing these trees to determine similarities or, equivalently, dissimilarities, becomes the fundamental issue. Typically, Tree Bisection and Reconnection (TBR) and Subtree Prune and Regraft (SPR) distances have been proposed to facilitate the comparison between different phylogenetic trees. In this paper, we give a survey on the aspects of computational complexity, fixed-parameter algorithms, and approximation algorithms for computing the TBR and SPR distances of phylogenetic trees.