We report and discuss nondiffractive and nondispersive spatiotemporal localized wave packets propagating in optical cubic nonlinear or Kerr media. In particular, we demonstrate the existence of novel families of dark spatiotemporal soliton solutions of the (2+1)D nonlinear Schrödinger equation (NLSE), derived from the hydrodynamic exact soliton dynamics of the (2+1)D shallow water Kadomtsev-Petviashvili (KP) model, inheriting their complex dynamics and interaction properties.