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This article is concerned with ergodic theory fortransformations which preserve an infinite measure. In the firstpart we present an overview of the invertible case with a focuson weakly wandering sequences and their applications to num-ber theory as it has developed over the last fifty years. Thesecond part presents a very preliminary investigation into ex-tending weakly wandering sequences to the non-invertible case.This consists primarily of a few examples which illustrate thecomplexities which arise in the non-invertible case.