This volume contains survey papers on the theory of operator algebras based on lectures given at the "Lluis Santalo" Summer School of the Real Sociedad Matematica Espanola, held in July 2008 at the Universidad Internacional Menendez Pelayo, in Santander (Spain). Topics in this volume cover current fundamental aspects of the theory of operator algebras, which have important applications such as: $K$-Theory, the Cuntz semigroup, and Classification for $C^*$-algebras Modular Theory for von Neumann algebras and applications to Quantum Field Theory Amenability, Hyperbolic Groups, and Operator Algebras. The theory of operator algebras, introduced in the thirties by J. von Neumann and F. J. Murray, was developed in close relationship with fundamental aspects of functional analysis, ergodic theory, harmonic analysis, and quantum physics. More recently, this field has shown many other fruitful interrelations with several areas of mathematics and mathematical physics.