  We develop exact Markov chain Monte Carlo methods for discretely-sampled,
directly and indirectly observed diffusions. The qualification "exact" refers
to the fact that the invariant and limiting distribution of the Markov chains
is the posterior distribution of the parameters free of any discretisation
error. The class of processes to which our methods directly apply are those
which can be simulated using the most general to date exact simulation
algorithm. The article introduces various methods to boost the performance of
the basic scheme, including reparametrisations and auxiliary Poisson sampling.
We contrast both theoretically and empirically how this new approach compares
to irreducible high frequency imputation, which is the state-of-the-art
alternative for the class of processes we consider, and we uncover intriguing
connections. All methods discussed in the article are tested on typical
examples.
