Subdivision is an efficient tool to explain curves and surfaces in geometric modeling and computer aided geometric design. Subdivision schemes are very helpful techniques to produced smooth curves and surfaces from finite set of control points. The aim of this dissertation is to introduce variety of subdivision schemes for curve and surface designing based on complexity, arity and parameter. Several simple and well-organized formulae are presented which generate the different kind of parametric and non-parametric subdivision schemes. Many well known existing schemes are generated by proposed formulae. Convergence and smoothness of curves and surfaces subdivision schemes are presented by using Laurent polynomial method. Shape preserving properties such as monotonicity, convexity and concavity preservation of data fitting are derived. Some of significant properties of proposed subdivision schemes such as Hölder regularity, polynomial generation, polynomial reproduction, approximation order and support of basic limit function are also discussed. Visual performances of the schemes have also been demonstrated through different examples.