Introduction -- Iteration of conformal maps -- On stochastic models and connection to statistical physics -- An example: percolation model and Cardy's formula -- On reading this book -- Introduction to stochastic calculus -- Brownian motion -- Stochastic integration -- Itô's formula -- Further topics in stochastic calculus -- Conformal invariance of two-dimensional Brownian motion -- Weak convergence of probability measures -- Introduction to conformal mappings -- Harmonic functions -- Conformal maps -- From Area theorem to distortion -- Conformally invariant tools -- Loewner equation -- Conformal maps of the upper half-plane -- Loewner chains -- Loewner equations in D and Sp -- Schramm-Loewner evolution.-Schramm-Loewner evolution and its elementary properties -- Advanced properties of SLE -- Proofs for some of the advanced properties -- Variants of SLE -- Moments of the derivative of the Loewner map of SLE(k) -- Regularity and convergence of random curves -- Continuity properties of the Loewner chains -- Continuity of SLE(k) -- Convergence of interfaces in the site percolation model -- Index.