  For a given set of $n\times n$ matrices $\mathcal F$, we study the union of
the $C$-numerical ranges of the matrices in the set $\mathcal F$, denoted by
$W_C({\mathcal F})$. We obtain basic algebraic and topological properties of
$W_C({\mathcal F})$, and show that there are connections between the geometric
properties of $W_C({\mathcal F})$ and the algebraic properties of $C$ and the
matrices in ${\mathcal F}$. Furthermore, we consider the starshapedness and
convexity of the set $W_C({\mathcal F})$. In particular, we show that if
${\mathcal F}$ is the convex hull of two matrices such that $W_C(A)$ and
$W_C(B)$ are convex, then the set $W_C({\mathcal F})$ is star-shaped. We also
investigate the extensions of the results to the joint $C$-numerical range of
an $m$-tuple of matrices.
