A space $X$ is {t almost countably compact\/} if for every countable open cover $\Cal U$ of $X$, there exists a finite subset $\Cal V$ of $\Cal U$ such that $\bigcup\{\overline{V}:V\in \Cal V\}=X$. In this paper, we investigate the relationship between almost countably compact spaces and countably compact spaces, and also study topological properties of almost countably compact spaces.