  In this paper, we systematically study jump and variational inequalities for
rough operators, whose research have been initiated by Jones {\it et al}. More
precisely, we show some jump and variational inequalities for the families
$\mathcal T:=\{T_\varepsilon\}_{\varepsilon>0}$ of truncated singular integrals
and $\mathcal M:=\{M_t\}_{t>0}$ of averaging operators with rough kernels,
which are defined respectively by $$ T_\varepsilon
f(x)=\int_{|y|>\varepsilon}\frac{\Omega(y')}{|y|^n}f(x-y)dy $$ and $$M_t
f(x)=\frac1{t^n}\int_{|y|<t}\Omega(y')f(x-y)dy, $$ where the kernel $\Omega$
belongs to $L\log^+\!\!L(\mathbf S^{n-1})$ or $H^1(\mathbf S^{n-1})$ or
$\mathcal{G}_\alpha(\mathbf S^{n-1})$ (the condition introduced by Grafakos and
Stefanov). Some of our results are sharp in the sense that the underlying
assumptions are the best known conditions for the boundedness of corresponding
maximal operators.
