Term rewriting is an appealing technique for performing program analysis and program transformation. Tree (term) traversal is frequently used but is not supported by standard term rewriting. We extend many-sorted, first-order term rewriting with traversal functions that automate tree traversal in a simple and type-safe way. Traversal functions can be bottom-up or top-down traversals and can either traverse all nodes in a tree or can stop the traversal at a certain depth as soon as a matching node is found. They can either define sort-preserving transformations or mappings to a fixed sort. We give small and somewhat larger examples of traversal functions and describe their operational semantics and implementation. An assessment of various applications and a discussion conclude the article.