Various types of infinite dimensionality of compact Hausdorff spaces are studied. In particular, it is shown that the classes of compact Hausdorff spaces for which the small and the large transfinite dimensions are defined coincide. An example, giving a negative solution of Aleksandrov's problem on the coincidence of countable dimensionality and weak infinite dimensionality in the class of compact Hausdorff spaces, is constructed. Bibliography: 19 titles.