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Mallows random permutations are random permutations with a slightly different probability measure than the uniform Haar measure. By choosing the parameters appropriately, however, one can choose the Mallows measures to be asymptotically absolutely continuous with respect to the uniform measures in a certain sense. We use the well known theorem of Vershik and Kerov, and Logan and Shepp, which gives a law of large numbers for the lengths of the longest increasing subsequences for uniform random permutations, to prove an analogous result for Mallows random permutations. The interesting problem of describing fluctuations is open (and was posed by Borodin, Diaconis and Fulman in a recent paper). This talk is based on joint work with Carl Mueller.