This talk is devoted to the topic of subdivision schemes, a special class of iterative methods for generating continuous curves and surfaces via the recursive application of suitable local refinement rules to a coarse initial set of prescribed control points. Due to their efficiency and simplicity of implementation, subdivision schemes are ones of the most used representation models in computer graphics and animation. Recently, they have shown their usefulness also in different areas of application like biomedical imaging and isogeometric analysis. Important tools for both the construction of linear subdivision schemes and the analysis of their properties are provided by classical numerical linear algebra techniques or adequate modifications of them. In particular, the construction of interpolatory subdivision schemes capable of generating curves and surfaces that pass through the initial set of prescribed control points, relies on algebraic strategies that differ according to the symmetry properties of the underlying refinement rules. The goal of this talk is to show some of the constructive strategies proposed in the literature for the subclass of stationary, odd- and even-symmetric, interpolatory subdivision schemes of arbitrary arity.