  The study of critical properties of systems with long-range interactions has
attracted in the last decades a continuing interest and motivated the
development of several analytical and numerical techniques, in particular in
connection with spin models. From the point of view of the investigation of
their criticality, a special role is played by systems in which the
interactions are long-range enough that their universality class is different
from the short-range case and, nevertheless, they maintain the extensivity of
thermodynamical quantities. Such interactions are often called weak long-range.
In this paper we focus on the study of the critical behaviour of spin systems
with weak-long range couplings using renormalization group, and we review their
remarkable properties. For the sake of clarity and self-consistency, we start
from the classical $O(N)$ spin models and we then move to quantum spin systems.
