Abstract: In the first chapter we consider critical curves - conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. The multifractal spectrum of harmonic measure and other fractal characteristics of critical curves are obtained. In the second chapter a stochastic growth process of branching trees based on stochastic Loewner evolution is introduced by adding a stable Levy process to the forcing. Its behavior is analyzed analytically and numerically. The properties of the growth change qualitatively and singularly at critical values of parameters.