Smoothing is an estimation method whereby a classical state (probability distribution for classical variables) at a given time is conditioned on all-time (both past and future) observations. This method has been well developed in classical systems, but its application to quantum systems has only recently begun to be explored. Previous works have used the term "quantum smoothing" to mean estimating classical parameters, (Phys. Rev. Lett., 102, 250403, (2009)), which is essentially classical smoothing in which the noisy observation of the classical parameters is mediated by a quantum system. In this thesis, I define a smoothed quantum state for a partially monitored open quantum system, conditioned on an all-time monitoring-derived record. I calculate the smoothed distribution for a hypothetical unobserved record Ut which, when added to the real record O, would complete the monitoring, yielding a pure-state "quantum trajectory". Averaging the pure state over this smoothed distribution yields the (mixed) smoothed quantum state S that is typically purer than the state F conditioned only on the past. I also study how the choice of actual unravelling affects the purity increment over that of the conventional (filtered) state conditioned only on the past record.