  L\'evy Flights are paradigmatic generalised random walk processes, in which
the independent stationary increments---the "jump lengths"---are drawn from an
$\alpha$-stable jump length distribution with long-tailed, power-law asymptote.
As a result, the variance of L\'evy Flights diverges and the trajectory is
characterised by occasional extremely long jumps. Such long jumps significantly
decrease the probability to revisit previous points of visitation, rendering
L\'evy Flights efficient search processes in one and two dimensions. To further
quantify their precise property as random search strategies we here study the
first-passage time properties of L\'evy Flights in one-dimensional
semi-infinite and bounded domains for symmetric and asymmetric jump length
distributions. To obtain the full probability density function of first-passage
times for these cases we employ two complementary methods. One approach is
based on the space-fractional diffusion equation for the probability density
function, from which the survival probability is obtained for different values
of the stable index $\alpha$ and the skewness (asymmetry) parameter $\beta$.
The other approach is based on the stochastic Langevin equation with
$\alpha$-stable driving noise. Both methods have their advantages and
disadvantages for explicit calculations and numerical evaluation, and the
complementary approach involving both methods will be profitable for concrete
applications. We also make use of the Skorokhod theorem for processes with
independent increments and demonstrate that the numerical results are in good
agreement with the analytical expressions for the probability density function
of the first-passage times.
