The results in the present paper show that the homogeneous locally compact $ANR$-spaces share many properties of Euclidean spaces. This is in accordance with the Bing-Borsuk conjecture \cite{bb} stating that every $n$-dimensional homogeneous metric $ANR$-compactum with $n\geq 3$ is an Euclidean manifold. We describe the local structure of homogeneous $ANR$-spaces. Using that description, we provide a positive solution of the problem whether every finite-dimensional homogeneous metric $ANR$-compactum $X$ is dimensionally full-valued, i.e. $\dim X\times Y=\dim X+\dim Y$ for any metric compactum $Y$.