A theory of real-time dependence of Raman scattering for a pulse-mode laser is developed within second-order perturbation theory and using the wavepacket terminology. The rate of spontaneous Raman emission with a pulse correctly reduces to the dynamical equivalent of the Kramers-Heisenberg-Dirac expression in the monochromatic limit. We apply the theory to continuum Raman scattering for short and long pulses and varying pulse carrier frequency. The rate of Raman emission as a function of time and pulse carrier frequency, from an initial ground vibrational state to various final vibrational states, is shown to be structureless for all pulses, and for pulses that are longer than the dissociation time the rate also rises and decays with the pulses. This is contrary to recent reports of recurring resonance fluorescence-type structures at long times after the pulse has vanished. We explain why such structures are unphysical for continuum Raman scattering. Results are also presented for excitation from an initial first excited vibrational state.