  The purpose of this paper is, as part of the stratification of Cohen-Macaulay
rings, to investigate the question of when the fiber products are almost
Gorenstein rings. We show that the fiber product $R \times_T S$ of
Cohen-Macaulay local rings $R$, $S$ of the same dimension $d>0$ over a regular
local ring $T$ with $\dim T=d-1$ is an almost Gorenstein ring if and only if so
are $R$ and $S$. Besides, the other generalizations of Gorenstein properties
are also explored.
