Permutations of [n] with all cycle lengths even, or with all cycle lengths odd, have on average about (log n)/2 cycles, compared to log n for permutations in general. Furthermore, random permutations from these two classes, and from permutations selected with weight proportional to 1/2 raised to the power of their number of cycles, have similar asymptotic propeties. These are examples of random _weighted permutations_, or permutations in which each cycle is endowed with a (multiplicative) weight which is a function fo the cycle size. I will present exact results on such permutations, with combinatorial proofs, in certain simple cases, and asymptotic results showing how statistical properties of random weighted permutations arise from statistical properties of the corresponding weighting sequence.