The lecture will attempt to present both the historical roots, as well as some of the principal developments and applications to other fields of Mathematics of this subject within the past three decades, underlying the role of Linear Algebra. It will concentrate on explicit presentations of representations of a given algebra, employing some related concepts such as classification problems (Brauer-Thrall conjectures, finite, tame and wild types), hereditary algebras and graphs, Coxeter functors and their linear transformations, as well as more recent results on quasi-hereditary algebras in relation to semi-simple complex Lie algebras. These concepts will be illustrated on solving some classification problems of geometry (von Staud's pairs and Kronecker's pairs of matrices), modular representations of groups (A4 over GF2) and C*-algebras (Jones index)..