  We consider the fiber cone of monomial ideals. It is shown that for monomial
ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber
cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth
for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which
are generated by $4$ elements. For these classes of ideals we also show that
$F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is
generated by at most 3 elements. In all the cases a minimal set of generators
of $J$ is determined.
