  We discuss two problems concerning the class Eberlein compacta, i.e., weakly
compact subspaces of Banach spaces. The first one deals with preservation of
some classes of scattered Eberlein compacta under continuous images. The second
one concerns the known problem of the existence of nonmetrizable compact spaces
without nonmetrizable zero-dimensional closed subspaces. We show that the
existence of such Eberlein compacta is consistent with ZFC. We also show that
it is consistent with ZFC that each Eberlein compact space of weight $>
\omega_1$ contains a nonmetrizable closed zero-dimensional subspace.
