Heterogeneous and distributed systems are nowadays increasingly being used to speed up the solution of the complex computational problems and to solve, so far, unsolvable ones. Today's trend shows that from the top 500 most powerful supercomputers in the world, 133 are already heterogeneous, that is, along with the traditional processors (CPUs) they are also equipped with accelerators, mostly graphical processing units (GPUs). It is expected that the future exascale computers, i.e. computers capable of performing more than 10^18 operations per second, will be based on the accelerators. Because of their high heterogeneity, the development of efficient and scalable algorithms and applications, achieving high utilization of such systems, is of the uttermost importance. The goal of this project is to develop new methods and to improve the existing algorithms of numerical linear algebra that will be capable of exploiting large heterogeneous systems while attaining very high performance. The main lines of the research are to increase the scalability of the algorithms (increased problem size and the number of computational resources), to reduce the communication overhead, and to exploit the complex memory hierarchies of the heterogeneous systems. Furthermore, new models for optimizing the parameters of algorithms will be explored concerning the architecture of the underlying computer system, to reduce the total execution time. Finally, the obtained research results will be applied in solving large, real-world computational problems arising in other fields of science, such as computational chemistry, material physics, and molecular medicine, and thus improve and accelerate their future research. The project will contribute to efficient parallelization and optimization of the algorithms for future heterogeneous and distributed systems, and will strengthen the research in the field of high-performance computing at the Ruder Boskovic Institute and the Republic of Croatia.