In this talk I will introduce quantum spin models and discuss the use of exact diagonalization technique to study their properties. The numerical methods applied to sparse matrices perform iterative diagonalization using Krylov eigensolvers. I will also discuss critical aspects of constructing the matrix in a reduced basis by considering lattice symmetries and an accelerated convergence in the diagonalization with additional sub-block diagonalization. The final part of the talk will be devoted to the discussion of solving problems with matrix sizes over a few hundred billion, which is relevant in the search of novel ground states on the Kagome lattice and other frustrated spin systems.