This is the abstract of the paper "Asy. Stability and Perron Cond. for Periodic Evol. Fam. " by C. Buse. For the whole paper send the command get evolve-l 98-00013 to: "listserv@rz.uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. Abstract: Extending earlier results of Datko, Pazy and Neerven on $C_0$-se\-mi\-groups, we prove the following. Let ${\cal U}$ be a $q$-periodic evolution family on a Banach space $X$ and $E$ be a normed function space over ${\bf R}_+$ with the property that $\lim\limits_{t\to\infty}|\chi_{[0, t]}|_E=\infty.$ If for any $x\in X,$ the map $t\mapsto ||U(t,0)x||:{\bf R}_+\to {\bf R}_+$ belongs to $E,$ then ${\cal U}$ is exponentially stable. On the other hand we prove that if $$\sup\limits_{t>0}\left\|\int\limits_0^tU(t, \xi)e^{i\mu\xi} f(\xi)d\xi\right\|= M(\mu, f)<\infty$$ for all $\mu\in{\bf R}$ and $f\in P_q({\bf R}_+, X),$ (that is $f$ is a $q$-periodic and continuous function on ${\bf R}_+$), then ${\cal U}$ is exponentially stable.