  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 a 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 letter, 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 than the corresponding
short-range one. 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 for $T<T_c$ and a disordered phase for $T>T_{\rm BKT}$.
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.
