Vector subdivision schemes and refinement equations have been studied extensively by functional analysts. However, it is rather unclear whether an arbitrary refinable function vector can be meaningfully applied to build free-form subdivision surfaces. In this talk, we shall first recall the basic idea of what it means by applying a subdivision scheme in the functional setting to the geometric setting. Afterwards I will talk about some new vector subdivision schemes: Hermite-typed (both interpolatory and non-interpolatory) and Lagrange-typed. We then illustrate how one can apply these schemes to derive algorithms for subdivision surfaces of arbitrary topology. Finally, some open problems will be discussed.