In the first part of the talk, we consider the question of when an  exponential martingale is a uniformly integrable martingale. A  classical  result of  Novikov states that for a continuous local martingale $M$, the  exponential  martingale  is a uniformly integrable martingale if $\exp(\frac{1}{2}[M]_\infty)$ is  integrable, and the constant $\frac{1}{2}$  is optimal. We observe that earlier  theorems by  Lépingle & Mémin yield Novikov-type results as corollaries, and  identify  optimal  constants in these criteria under certain conditions on the jumps. We also give elementary proofs of slight extensions of the Novikov-type criteria obtained  earlier. Finally, we show how to apply exponential martingales to the construction of various types of counting processes with stochastic intensities, for  example intensities depending on a diffusion.