  This paper is concerned with a class of nonlocal dispersive models -- the
$\theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation,
{\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: $$
(1-\partial_x^2)u_t+(1-\theta\partial_x^2)(\frac{u^2}{2})_x
=(1-4\theta)(\frac{u_x^2}{2})_x, $$ including integrable equations such as the
Camassa-Holm equation, $\theta=1/3$, and the Degasperis-Procesi equation,
$\theta=1/4$, as special models. We investigate both global regularity of
solutions and wave breaking phenomena for $\theta \in \mathbb{R}$. It is shown
that as $\theta$ increases regularity of solutions improves: (i) $0 <\theta <
1/4$, the solution will blow up when the momentum of initial data satisfies
certain sign conditions; (ii) $1/4 \leq \theta < 1/2$, the solution will blow
up when the slope of initial data is negative at one point; (iii) ${1/2} \leq
\theta \leq 1$ and $\theta=\frac{2n}{2n-1}, n\in \mathbb{N}$, global existence
of strong solutions is ensured. Moreover, if the momentum of initial data has a
definite sign, then for any $\theta\in \mathbb{R}$ global smoothness of the
corresponding solution is proved. Proofs are either based on the use of some
global invariants or based on exploration of favorable sign conditions of
quantities involving solution derivatives. Existence and uniqueness results of
global weak solutions for any $\theta \in \mathbb{R}$ are also presented. For
some restricted range of parameters results here are equivalent to those known
for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up
phenomena, and global solutions for the b-equation, {\it J. reine angew.
Math.}, {\bf 624} (2008)51--80.]
