  We investigate the first-passage dynamics of symmetric and asymmetric L\'evy
flights in a semi-infinite and bounded intervals. By solving the
space-fractional diffusion equation, we analyse the fractional-order moments of
the first-passage time probability density function for different values of the
index of stability and the skewness parameter. A comparison with results using
the Langevin approach to L\'evy flights is presented. For the semi-infinite
domain, in certain special cases analytic results are derived explicitly, and
in bounded intervals a general analytical expression for the mean first-passage
time of L\'evy flights with arbitrary skewness is presented. These results are
complemented with extensive numerical analyses.
