The growth dynamics for metallic filaments in conductive-bridge resistive-switching random access memory (CBRAM) are studied using the kinetic Monte Carlo (KMC) method. The physical process at the atomistic level is revealed in explaining the experimental observation that filament growth can originate at either the cathode or the anode. The statistical nature of the filament growth is best shown by the random topography of dendrite-like conductive paths obtained. Critical material properties, such as charged-particle mobility in the switching layer of a solid electrolyte or a dielectric, are mapped to KMC model parameters through activation energy, etc. The accuracy of the simulator is established by the good agreement between the simulated forming time and the measured data.