It is well-known that tree automata define exactly regular languages of trees. However for some problems one sometimes needs to test for equalities and disequalities of subtrees. For instance, ranges of non-linear tree transducers cannot be represented by tree automata. To overcome this problem some extensions of tree automata with tree (dis)equality constraints have been proposed. This talk surveys some of the most important models and their applications. Two families of automata are presented. First, we consider automata with local constraints. They have been used to solve problems related to pattern-matching, tree rewriting and more recently the tree homomorphism problem. Then, motivated by applications to XML processing and security protocols verification, we present a more recent model of tree automata with global constraints.