We define a general notion of locally standard $(Z_2)^m$-actions on
n-dimensional closed manifolds for all m>0. And we will give a standard way to
recover all such actions from the orbit space with some associated
characteristic functions. Then we will discuss the classification of closed
n-manifolds with locally standard $(Z_2)^m$-actions up to (weak) equivariant
homeomorphisms and some related topological problems.