In this thesis we introduce a framework for parallel MCMC methods which we call parallel adaptive importance sampling (PAIS). At each iteration we have an ensemble of particles, from which PAIS builds a kernel density estimate (KDE). We propose a new ensemble, using this KDE, that is weighted according to standard importance sampling rules. A state-of-the art resampling method from the optimal transportation literature, or alternatively our own novel resampling algorithm, can be used to produce an equally weighted ensemble from this weighted ensemble. This equally weighted ensemble is approximately distributed according to the target distribution and is used to progress the algorithm. The PAIS algorithm outputs a weighted sample. We introduce an adaptive scheme for PAIS which automatically tunes the scaling parameters required for efficient sampling. This adaptive tuning converges rapidly for the target distributions we have experimented with and significantly reduces the burn-in period of the algorithm. PAIS has been designed to work well on computers with parallel processing units available, and we have demonstrated that a doubling of the number of processing units available more than halves the number of iterations required to reach the same accuracy. The numerical examples have been implemented on a shared memory system. PAIS is incredibly flexible in terms of the proposal distributions and resampling methods we can use. Throughout the thesis we introduce a number of these proposal schemes, and highlight when they may be of use. Of particular interest is the transport map based proposal scheme introduced in Chapter 7 which, while more expensive than the other schemes, allows us to sample efficiently from a wide range of complex target distributions.