Elementary algebra, the study of equations and polynomials, has inspired the development of modern algebraic structures, as groups, fields, rings and Lie-algebras. Linear algebra, the theory of vector spaces and linear transformations, is a central topic within most parts of basic and applied mathematics. A group, ring or Lie-algebra can often be investigated by interpreting its elements as linear transformations on vector spaces, and in this way e.g. complicated non-linear physical problems can be made m ore concrete and handled by methods of linear algebra. This is the idea of representation theory. The main aim of this project is to study the interplay between representation theory and combinatorics. Combinatorial techniques has for a long time been a part of representation theory. One of the reasons for this is that non-commutative, non-linear structures often can be understood in terms of so called quivers or directed graphs. Several specific applications of combinatorial techniques within represent ation theory will be investigated as part of the project. It is also an important part of the project to further develop computer software in order to implement some of these combinatorial methods. Another aim of the project is to use modern techniques w ithin representation theory to study concrete combinatorial problems. In particular, mutation of quivers is a central topic. This is a combinatorial phenomenon, which for the last 10-15 years has proved to be central in many part of mathematics, often as part of a structure called cluster algebras. Mutation of quivers is also linked to mathematical physics, in particular to string theory and mirror symmetry. Applications of homological techniques within representation theory, often formulated in terms of triangulated categories, has turned out to be particularly fruitful in the study of mutation of quivers and cluster algebras. One of the main aims of the project is to give further contributions to this theory.