The paper contains many characterizations of quasicontinuity of multifunctions and relationships between quasicontinuity and somewhat continuity. The most important result states that the set of points at which a multifunction F: \(X\to Y\) is simultaneously lower and upper quasicontinuous, but fails to be quasicontinuous, is of the first category, provided Y is second countable and F has compact values. Quasicontinuity of multifunction defined on product spaces, is established under suitable conditions imposed on sections of F (e.g. quasicontinuity, lower and upper quasicontinuity and somewhat continuity of horizontal sections and somewhat continuity of vertical sections, etc.)] Finally, it is shown that the Vietoris topology leads to a more restrictive notion of quasicontinuity, than the usual notion introduced by V. Popa and investigated by T. Neubrúnn.