  We introduce the category of {\it locally $k$-standard $T$-manifolds} which
includes well-known classes of manifolds such as toric and quasitoric
manifolds, good contact toric manifolds and moment-angle manifolds. They are
smooth manifolds with well-behaved actions of tori. We study their topological
properties, such as fundamental groups and equivariant cohomology algebras.
Then, we discuss when the torus equivariant cohomology algebra distinguishes
them up to weakly equivariant homeomorphism.
