  The geometric and algebraic properties of smooth projective varieties with
1-regular structure sheaf are well understood, and the complete classification
of these varieties is a classical result. The aim of this paper is to study the
next case: smooth projective varieties with 2-regular structure sheaf. First,
we give a classification of such varieties using adjunction mappings. Next,
under suitable conditions, we study the syzygies of section rings of those
varieties to understand the structure of the Betti tables, and show a sharp
bound for Castelnuovo-Mumford regularity.
