  Microswimmers are microscopic active agents capable of harvesting energy from
the surrounding environment and converting it into self-propulsion and directed
motion. This peculiar feature characterizes them as out-of-equilibrium systems
that break microscopic reversibility. The problem of finding a specific target
in a complex environment is essential for these agents since it is employed for
a variety of purposes, from foraging nourishment to escaping potential threats.
Here, we provide a detailed study of the target search process for
microswimmers exploring complex environments. To this end, we generalize
Transition Path Theory, the rigorous statistical mechanics description of
transition processes, to the target-search problem. One of the main results of
this thesis is the generalization to non-equilibrium systems of the Transition
Path Sampling (TPS) algorithm, which was originally designed to simulate rare
transitions in passive systems. The TPS algorithm relies on microscopic
reversibility for its functioning, therefore its generalization to
out-of-equilibrium systems lacking detailed balance and microscopic
reversibility has remained a major challenge. Within this work, we generalize
the TPS algorithm to the case of an active Brownian particle, i.e. a
paradigmatic model for microswimmers, and we obtain a first insight into the
counterintuitive target-search pathways explored by these agents. The second
result of this thesis is a systematic characterization of the target-search
path ensemble for an active particle exploring an energy landscape. The third
and final original contribution of this Ph.D. thesis is the generalization of
the concept of the committor function to target-search problems, with a
validation of our theory against experiments of a camphor self-propelled disk.
