  We study random walks on groups with the feature that, roughly speaking,
successive positions of the walk tend to be "aligned". We formalize and
quantify this property by means of the notion of deviation inequalities. We
show that deviation inequalities have several consequences including Central
Limit Theorems, the local Lipschitz continuity of the rate of escape and
entropy, as well as linear upper and lower bounds on the variance of the
distance of the position of the walk from its initial point. In a second part
of the paper, we show that the (exponential) deviation inequality holds for
measures with exponential tail on acylindrically hyperbolic groups. These
include non-elementary (relatively) hyperbolic groups, Mapping Class Groups,
many groups acting on CAT(0) spaces and small cancellation groups.
