Schramm-Loewner evolutions (SLE) are random, conformally invariant curves that describe continuum limits of interfaces in 2-d statistical physics at criticality. After a brief introduction to SLEs, I will discuss an approach to SLEs with quantum physics flavour, \`a la Bauer and Bernard. A closer study of this approach reveals natural appearance of many kinds of representations of Virasoro algebra, not only highest weight representations. To illustrate what this algebraic point of view teaches us about SLEs, I'll show how to obtain partial results of two well-known SLE questions: reversibility of chordal SLE trace and ``Duplantier duality'' for SLEs.