This book grew from a series of lectures given to graduate students of condenses matter physics. This introduction to conformal invariance and to two-dimensional critical phenomena reflects their theoretical background. The algebraic foundation of critical theories in two dimensions is introduced and explained as is the role of modular invariance for the partition function. A large part of the book is devoted to numerical methods and their application in various models. Finite-size scaling techniques and their conformal extensions are treated in detail. Transfer matrix diagonalization methods are used to study, among others, the Ising and Potts models, including tricritical behaviour, the Ashkin-Teller model systems with continuous symmetries such as the XY model and the XXZ quantum chain as well as the Yang-Lee edge singularity. Numerical methods also make it possible to describe the vicinity of the critical point. The exact S-matrix approach, truncation method, the thermodynamic Bethe ansatz and the asymptotic finite-size scaling function technique are illustrated on simple models. The integrability of the two-dimensional Ising model in a magnetic field is dealt with. Finally, the extension of conformal invariance appropriate to the study of surface critical phenomena and defect line problems is described, and the book closes with an outlook towards possible applications in critical dynamics.