  Castelnuovo-Mumford regularity is an important invariant of projective
algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a
bound for regularity in terms of the codimension and degree,i.e.,
Castelnuovo-Mumford regularity of a given variety $X$ is less than or equal to
$deg(X)-codim(X)+1$. This regularity conjecture (including classification of
examples on the boundary) was verified for integral curves (Castelnuovo,
Gruson, Lazarsfeld and Peskine), and for smooth surfaces (Pinkham, Lazarsfeld).
In this paper we prove that $reg(X) \le deg(X)-1$ for smooth threefolds $X$ in
P^5 and that the only varieties on the boundary are the Segre threefold and the
complete intersection of two quadrics. Furthermore, every smooth threefold $X$
in P^5 is $k$-normal for all $k \ge deg(X)-4$, which is the optimal bound as
the Palatini 3-fold of degree 7 shows.
