  We give a selection of major open problems involving selective properties,
diagonalizations, and covering properties for sets of real numbers.
  This is a revision of the version published as a chapter in the book
\textbf{Open Problems in Topology II} (E. Pearl, ed.), Elsevier B.V., 2007,
91--108. The present version reports solutions of some problems, uses
up-to-date notation, and update bibliography.
  Comments and further updates would be appreciated.
