The notion of quasi-particles is essential for understanding the behaviour of
complex many-body systems. A prototypical example of a quasi-particle, a
polaron, is an impurity strongly interacting with a surrounding medium. Fermi
polarons, created in a Fermi sea, provide a paradigmatic realization of this
concept. As an inherent and important property such quasi-particles interact
with each other via modulation of the medium. While quantum simulation
experiments with ultracold atoms have significantly improved our understanding
of individual polarons, the detection of their interactions has remained
elusive in these systems. Here, we report the unambiguous observation of
mediated interactions between Fermi polarons consisting of K impurities
embedded in a Fermi sea of Li atoms. Our results confirm two landmark
predictions of Landau's Fermi-liquid theory: the shift of the polaron energy
due to mediated interactions, linear in the concentration of impurities, and
its sign inversion with impurity quantum statistics. For weak to moderate
interactions between the impurities and the medium, we find excellent agreement
with the static (zero-momentum and energy) predictions of Fermi-liquid theory.
For stronger impurity-medium interactions, we show that the observed behaviour
at negative energies can be explained by a more refined many-body treatment
including retardation and molecule formatio