Partitions of [n] = {1,2,...,n} into sets of lists (A000262) are somewhat less numerous than partitions of [n] into lists of sets (A000670). Here we observe that the former are actually equinumerous with partitions of [n] into lists of noncrossing sets and give a bijective proof. We show that partitions of [n] into sets of noncrossing lists are counted by A088368 and generalize this result to introduce a transform on integer sequences that we dub the "noncrossing partition" transform. We also derive recurrence relations to count partitions of [n] into lists of noncrossing lists.