This book is an introduction to the most important clases of non-associative algebras, namely Lie, Jordan and alternative algebras. It also presents the different definitions of representation and bi-representation, starting with the classic concept for groups and ending with Eilenberg definition for general non-associative algebras. It focuses in two issues that can be solved with this theory, the existence of an adjoint bi-representation and the existence of a faithful bi-representation. In order to solve this two problems it introduces the theory of free algebras and polynomial identities, together with its main results.