Top-context-free tree languages (called {\em cor\'{e}gulier} by Arnold and Dauchet [A.Arnold, M.Dauchet, Transductions de For\^ets Reconnaissables Monadiques. for\^ets Cor\'eguli\`eres, RAIRO Informatique Th\'eorique et applications, 10(3):5--28, 1976; A.Arnold, M.Dauchet, Un Th\'eor\`eme de Duplication pour les For\^ets Alg\'ebriques, Journal of Computer and System Sciences, 13:223--244, 1976] constitute a natural subclass of context-free tree languages. In this paper, we give further evidence for the importance of this class by exhibiting certain closure properties. We systematically treat closure w.r.t.\ the operations {\em replacement} and {\em substitution} as well as w.r.t.\ the corresponding {\em iteration} operations; several other well-known language classes are considered as well. Furthermore, various characterizations of the regular top-context-free languages are given, among others by means of a restricted form of regular expressions.