Russell's paradox is as follows: Consider the set of all sets which are not members of themselves. Is this set a member of itself? If it is, then it is not. If it is not, then it is. A contradiction ensues.
Russell's paradox is a consequence of the axiom of comprehension and the existence of a universal set. Hence, in ZF set theory, the universal set does not exist. Alternatively, one could avoid Russell's paradox and retain a universal set by rejecting the axiom of comprehension.