  We wish to characterise when a L\'{e}vy process $X_t$ crosses boundaries like
$t^\kappa$, $\kappa>0$, in a one or two-sided sense, for small times $t$; thus,
we enquire when $\limsup_{t\downarrow 0}|X_t|/t^{\kappa}$,
$\limsup_{t\downarrow 0}X_t/t^{\kappa}$ and/or $\liminf_{t\downarrow
0}X_t/t^{\kappa}$ are almost surely (a.s.) finite or infinite. Necessary and
sufficient conditions are given for these possibilities for all values of
$\kappa>0$. Often (for many values of $\kappa$), when the limsups are finite
a.s., they are in fact zero, as we show, but the limsups may in some
circumstances take finite, nonzero, values, a.s. In general, the process
crosses one or two-sided boundaries in quite different ways, but surprisingly
this is not so for the case $\kappa=1/2$. An integral test is given to
distinguish the possibilities in that case. Some results relating to other
norming sequences for $X$, and when $X$ is centered at a nonstochastic
function, are also given.
