  For a L\'evy process on the real line, we provide complete criteria for the
finiteness of exponential moments of the first passage time into the interval
$(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit
time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we
derive their respective asymptotic behavior as $r \to \infty$.
