This is the abstract of the paper "New Spectral Criteria for Almost Periodic Solutions of Evolution Equations" by T. Naito and Nguyen Van Minh and Jong Son Shin. For the whole paper send the command get evolve-l 99-00006 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: We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form $\dot{x}=A(t)x+f(t) \ (*)$, with $f$ having precompact range, which will be then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonnant case where $\overline{e^{isp(f)}}$ may intersect the spectrum of the monodromy operator $P$ of $(*)$ (here $sp(f)$ denotes the Carleman spectrum of $f$). We show that if $(*)$ has a bounded uniformly continuous mild solution $u$ and $\sigma _\Gamma (P)\backslash \overline{e^{isp(f)}}$ is closed, where $\sigma _\Gamma (P)$ denotes the part of $\sigma (P)$ on the unit circle, then $(*)$ has a bounded uniformly continuous mild solution $w$ such that $\overline{e^{isp(w)}} =\overline{e^{isp(f)}}$. Moreover, $w$ is a "spectral component" of $u$. This allows to solve the general Massera-typed problem for almost periodic solutions. Various spectral criteria for the existence of almost periodic, quasi-periodic mild solutions to $(*)$ are given.