We present a numerical study of the domain growth kinetics of the continuous, two-dimensional, q-state Potts model with a nonconserved order parameter. After the system is quenched from a high-temperature disordered state to a finite temperature in the ordered region of the phase diagram, the development of ordered domains in time is analyzed by direct numerical solution of the associated Langevin equations. We find that the domain growth in the model, after a short initial transient time, is well described by a characteristic length L(t) which increases with time as L(t)tn. The kinetic exponent n is determined to be (1/2 for both q=4 and 15. The quasistatic structure factor S(k,t) obeys dynamical scaling during the growth process, as expected. Our results are in agreement with those obtained from recent Monte Carlo studies of the discrete Potts model with stochastic dynamics, and indicate that the domain growth kinetics of the two-dimensional q-state Potts model is characterized by a Lifshitz-Allen-Cahn kinetic exponent n=(1/2, independent of q for a nonconserved order parameter.