V and I band CCD images of the SMC cluster NGC 330 have been obtained over an interval of 4 years with the aim of identifying variable stars in and around this rich, young cluster. The search has revealed 22 Cepheid variables, of which 14 are newly discovered. All of these appear to belong to the SMC field population rather than being cluster members. Twenty long-period variables, and nine other variables were identified including two eclipsing variables (one newly discovered), and four close binary stars, two of which show emission lines. The Cepheids indicate that only small to moderate amounts of convective core overshoot (f_ov_<~0.25) occur during the main-sequence phases for stars of mass 2-3M_{sun}_. The Cepheid period-luminosity-color relation obtained from VI photometry, when compared to the theoretical relation of Chiosi, et al. (1993ApJS...86..541C), yields a distance modulus for the short-period Cepheids observed during this study of 18.92 if no overshoot is assumed, or 18.72 if moderate overshoot (f_ov=0.25) is adopted. Only two modes of pulsation are present in this group of Cepheids, most likely they are the fundamental and first overtone. Addresses: Sebo K.M. Wood P.R. Mount Stromlo and Siding Spring Observatories, Private Bag, Weston Creek PO, A.C.T. 2611 Australia