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We study the (metric) Diophantine approximation properties of uniformly expanding transformations and some non-uniformly expanding transformations, i.e. transformations T (x) with an associated countable (not necessarily finite) partition and a return time function R(x) (constant on the blocks of the partition) so that (T) over cap (x) = T-R(x) (x) is uniformly expanding, and we obtain Borel-Cantelli results on hitting times of shrinking targets. Our arguments do not require the so-called big image property for (T) over cap and our results contain most of the diversity of examples of slowly mixing systems. We also obtain, with related techniques, results for one-sided topological Markov chains over a countable alphabet with a Gibbs measure.