  The Berezinskii-Kostelitz-Thouless (BKT) transition is the paradigmatic
example of a topological phase transition without symmetry-breaking, where a
quasi-ordered phase, characterized by the power law scaling of the correlation
functions at low temperature, is disrupted by the proliferation of topological
excitations above the critical temperature $T_{\rm BKT}$. In this paper, we
consider the effect of long-range decaying couplings $\sim r^{-2-\sigma}$ on
this phenomenon. After pointing out the relevance of this non-trivial problem,
we discuss the phase diagram, which is far richer with respect to the
short-range interactions case. It features -- for $7/4<\sigma<2$ -- a quasi
ordered phase in a finite temperature range $T_c < T < T_{\rm BKT}$, which
occurs between a symmetry broken phase at $T<T_c$ and the disordered phase at
high-temperatures $T>T_{\rm BKT}$. The transition temperature $T_c$ displays
unique universal features quite different from those of the traditional,
short-range XY model. Given the universal nature of our findings, they may be
observed in current experimental realizations in $2D$ atomic, molecular and
optical quantum systems.
