Abstract: A bounded linear operator $T$ on Banach space is subspace-supercyclic for a nonzero subspace $M$ if there exists a vector whose projective orbit intersects the subspace $M$ in a relatively dense set. We constructed examples to show that subspace-supercyclic is not a strictly infinite dimensional
phenomenon, and that some subspace-supercyclic operators are not supercyclic. We provided a subspace-supercyclicity criterion and offered two necessary and sufficient conditions for a path of bounded linear operators to have a dense $G_\delta$ set of common subspace-hypercyclic vectors and common subspace-supercyclic vectors.

Key words: subspace-supercyclicity, common subspace-hypercyclic vectors, common subspace-supercyclic vectors