  We obtain some new upper bounds on the Ramsey numbers of the form
$R(\underbrace{C_4,\ldots,C_4}_m,G_1,\ldots,G_n)$, where $m\ge 1$ and
$G_1,\ldots,G_n$ are arbitrary graphs. We focus on the cases of $G_i$'s being
complete, star $K_{1,k}$ or book graphs $B_k$, where $B_k=K_2+kK_1$. If $k\ge
2$, then our main upper bound theorem implies that $$R(C_4,B_k) \le
R(C_4,K_{1,k})+\left\lceil\sqrt{R(C_4,K_{1,k})}\right\rceil+1.$$
  Our techniques are used to obtain new upper bounds in several concrete cases,
including: $R(C_4,K_{11})\leq 43$, $R(C_4,K_{12})\leq 51$, $R(C_4,K_3,K_4)\leq
29$, $R(C_4, K_4,K_4)\leq 66$, $R(C_4,K_3,K_3,K_3)\leq 57$,
$R(C_4,C_4,K_3,K_4)\leq 75$, and $R(C_4,C_4,K_4,K_4)\leq 177$, and also
$R(C_4,B_{17})\leq 28$.
