In this talk, we will discuss the role of parafermionic observables in the study of several planar statistical physics models. These objects have been introduced recently by Smirnov and Cardy and have been instrumental in Smirnov's proof of conformal invariance of the Ising model. We will explain how they can be combined with combinatorial and probabilistic arguments to compute the connective constant for self-avoiding walks (the n=0 loop O(n)-model) on the hexagonal lattice, and to provide information on the critical phase of the Fortuin-Kasteleyn percolation (a graphical representation of Potts models). As an application of their use for FK percolation, we will show the absence of spontaneous magnetization for the critical planar Potts models with 2, 3 and 4 colors, thus proving part of the conjecture asserting that the planar Potts models undergo a discontinuous phase transition if and only if the number of colors is greater than 4.