The aim of this thesis is to study the convergence and smoothness of certain nonlinear interpolatory curve subdivision schemes. The emphasis will be on the iterated geometric schemes, which are extensions of the nonlinear four-point scheme by Dyn, Floater and Hormann, based on iterated chordal and centripetal parameterizations. Dyn et al. show convergence of the scheme for uniform, centripetal and chordal parameterizations, i.e. alpha=0,1/2,1, but we here consider the entire interval [0,1] of alpha, and derive new results concerning convergence. In particular, we show that the scheme by Dyn et al. is C^0 for all alpha in [1/2,1], but that there always exist control points such that the limit curve is not well defined for all alpha in (0,1/2). We also show that a scheme based on the iterated geometric schemes and the six-point scheme with tension parameter, is C^0 for a range of parameters. The aforementioned schemes are then shown to fit into a recent framework by Ewald et al., for studying smoothness criteria, and we propose modified refinement rules based on the circle preserving scheme by Sabin and Dodgson to better fit this framework. Lastly, numerical experiments are carried out to measure the smoothness of the schemes, and a new way to generate the multilevel grid based on the geometry of the points, is proposed.