A bounded operator $T$ acting on a Banach space $\cal B$ is said to be supercyclic if there is a vector $x\,{\in}\,\cal B$ such that the projective orbit $\{\lambda T^n x\,{:}\,n\,{\geq}\,0\text{ and } \lambda\,{\in}\,\Bbb C \}$ is dense in $\cal B$. Examples of supercyclic operators are hypercyclic operators, in which the orbit itself is dense without the help of scalar multiples. Supercyclic operators are, in turn, a special case of cyclic operators. An operator is called cyclic if the linear span of the orbit of some vector is dense in the underlying space. This survey describes some recent results on linear subspaces in which all elements, except the zero vector, are supercyclic for a given supercyclic operator.