  The paper deals with two issues: the existence of universal models of a
theory T and related properties when cardinal arithmetic does not give this
existence offhand. In the first section we prove that simple theories (e.g.,
theories without the tree property, a class properly containing the stable
theories) behaves ``better'' than theories with the strict order property, by
criterion from [Sh:457]. In the second section we introduce properties SOP_n
such that the strict order property implies SOP_{n+1}, which implies SOP_n,
which in turn implies the tree property. Now SOP_4 already implies
non-existence of universal models in cases where earlier the strict order
property was needed, and SOP_3 implies maximality in the Keisler order, again
improving an earlier result which had used the strict order property.
