  We study $h$-vectors of simplicial complexes which satisfy Serre's condition
($S_r$). We say that a simplicial complex $\Delta$ satisfies Serre's condition
($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all faces $F \in \Delta$ and for
all $i < \min \{r-1,\dim \lk_\Delta(F)\}$, where $\lk_\Delta(F)$ is the link of
$\Delta$ with respect to $F$ and where $\tilde H_i(\Delta;K)$ is the reduced
homology groups of $\Delta$ over a field $K$. The main result of this paper is
that if $\Delta$ satisfies Serre's condition ($S_r$) then (i) $h_k(\Delta)$ is
non-negative for $k =0,1,...,r$ and (ii) $\sum_{k\geq r}h_k(\Delta)$ is
non-negative.
