  In this chapter, we consider the problem of a non-Markovian random walker
(displaying memory effects) searching for a target. We review an approach that
links the first passage statistics to the properties of trajectories followed
by the random walker in the future of the first passage time. This approach
holds in one and higher spatial dimensions, when the dynamics in the vicinity
of the target is Gaussian, and it is applied to three paradigmatic target
search problems: the search for a target in confinement, the search for a
rarely reached configuration (rare event kinetics), or the search for a target
in infinite space, for processes featuring stationary increments or transient
aging. The theory gives access to the mean first passage time (when it exists)
or to the behavior of the survival probability at long times, and agrees with
the available exact results obtained perturbatively for examples of weakly
non-Markovian processes. This general approach reveals that the
characterization of the non-equilibrium state of the system at the instant of
first passage is key to derive first-passage kinetics, and provides a new
methodology, via the analysis of trajectories after the first-passage, to make
it quantitative.
