Recently, Babson and Steingr\'{\i}msson have introduced generalised
permutation patterns that allow the requirement that two adjacent
letters in a pattern must be adjacent in the permutation. We
consider pattern avoidance for such patterns, and give a complete
solution for the number of permutations avoiding any single pattern
of length three with exactly one adjacent pair of letters. We also
give some results for the number of permutations avoiding two
different patterns. Relations are exhibited to several well studied
combinatorial structures, such as set partitions, Dyck paths,
Motzkin paths, and involutions. Furthermore, a new class of set
partitions, called monotone partitions, is defined and shown to be
in one-to-one correspondence with non-overlapping partitions.