  Ascent sequences are sequences of nonnegative integers with restrictions on
the size of each letter, depending on the number of ascents preceding it in the
sequence. Ascent sequences have recently been related to (2+2)-free posets and
various other combinatorial structures. We study pattern avoidance in ascent
sequences, giving several results for patterns of lengths up to 4, for Wilf
equivalence and for growth rates. We establish bijective connections between
pattern avoiding ascent sequences and various other combinatorial objects, in
particular with set partitions. We also make a number of conjectures related to
all of these aspects.
