  For two graphs $G,H$ and a positive integer $k$, the \emph{Gallai-Ramsey
number} $\operatorname{gr}_k(G,H)$ is defined as the minimum number of vertices
$n$ such that any $k$-edge-coloring of $K_n$ contains either a rainbow (all
different colored) copy of $G$ or a monochromatic copy of $H$. If $G$ and $H$
are both complete graphs, then we call it Gallai-Ramsey function. Fox and
Sudakov proved $\operatorname{gr}_k(K_s,K_t)\leq s^{4kt}$. Alon et al. showed
that $\operatorname{gr}_k(K_s,K_t)\leq (2s^3+4s^2)^{kt}$. In this paper, we
prove that $\operatorname{gr}_k(K_s,K_t)\leq 2^{kt}s^{3kt}$ for $t\geq 47$. We
also give better upper bounds for $\operatorname{gr}_k(G,H)$ when $G,H$ are
some special graphs. In this paper, we derive some lower bounds for
Gallai-Ramsey functions and numbers by Lov\'{a}sz Local Lemma.
