  In this paper, we consider the Cauchy problem for the $b$-equation. Firstly,
for $s>\frac32,$ if $u_{0}(x)\in H^{s}(\mathbb{R})$ and
$m_{0}(x)=u_{0}(x)-u_{0xx}(x)\in L^{1}(\mathbb{R}),$ the global solutions of
the $b$-equation is established when $b\geq1$ or $b\leq1.$ It's worth noting
that our global result is a new result which doesn't need the condition that
$m_{0}(x)$ keeps its sign. For $s<\frac32,$ it is shown (see [13]) that the
Cauchy problem of the $b$-equation is ill-posed in Sobolev space
$H^{s}(\mathbb{R})$ when $b>1$ or $b<1.$ In the present paper, for $s=\frac32,$
we prove that the Cauchy problem of the $b$-equation is also ill-posed in
$H^{\frac32}(\mathbb{R})$ in the sense of norm inflation by constructing a
class of special initial data when $b\neq1.$
