Abstract: In this article, two results concerning the periodic points and normality of meromorphic functions are obtained: (i) the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by letting $R(z)$ be a non-polynomial rational function, and if all zeros and poles of $R(z)-z$ are multiple, then $R^k(z)$ has at least $k+1$ fixed points in the complex plane for each integer $k\ge 2$; (ii) a complete solution to the problem of normality of meromorphic functions with periodic points is given by letting $\mathcal{F}$ be a family of meromorphic functions in a domain $D$, and letting $k\ge 2$ be a positive integer. If, for each $f\in \mathcal{F}$, all zeros and poles of $f(z)-z$ are multiple, and its iteration $f^k$ has at most $k$ distinct fixed points in $D$, then $\mathcal{F}$ is normal in $D$. Examples show that all of the conditions are the best possible.

Key words: normality, iteration, periodic points