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We study generalizations to division rings of the notions of algebraic and separable closures of fields, and of algebraically closed and separably closed fields. For example, we show that every division ring has at least one “algebraic closure” but also that there are division rings  H  with infinitely many “algebraic closures” which are pairwise non- H -isomorphic. We also produce division rings which contain, which are “separably closed” but which are not “algebraically closed”. Résumé Nous étudions des généralisations à la situation des corps non nécessairement commutatifs des notions de clôtures algébriques et séparables de corps commutatifs, et de corps commutatifs algébriquement clos et séparablement clos. Nous montrons, par exemple, que n’importe quel corps admet au moins une “clôture algébrique” mais aussi qu’il existe des corps  H  admettant une infinité de “clôtures algébriques” deux à deux non  H -isomorphes. Nous construisons également des corps contenant, “séparablement clos” mais non “algébriquement clos”.

Extensions algébriques maximales de corps gauches