Wavelets and subdivision have developed into important mathematical tools in such applications as signal analysis, digital image processing, computer-aided geometric design (CAGD), etc. Underlying the analysis of both wavelets and subdivision schemes are the issues of existence, uniqueness, regularity (or smoothness) and numerical evaluation of a refinable function. In fact, any wavelet or subdivision limit function is expressible as a linear combination of the integer shifts of some refinable function. Consequently, the wavelet or the limit function naturally inherits the properties of the associated refinable function. In particular, the regularity of the refinable function is preserved by both the wavelet as well as the subdivision limit function. Hence, the regularity of a refinable function substantially influences the efficiency of both the associated wavelet decomposition algorithm and the corresponding subdivision scheme. This work investigates sufficient conditions that guarantee the global regularity of a given refinable function. It will appeal to both academics and practitioners.