Quasiseparable matrices is a relatively new class of rank-structured matrices that naturally arise in many applications of numerical analysis. Examples include interpolation by polynomials, factorization of moment matrices, polynomial root finding. We will talk specifically about their application in solving partial differential equations and optimal control problems. We will show that for a certain class of problems the use of quasiseparable matrices techniques leads to system solvers of asymptotically linear complexity in the dimension of the system. The talk aims to give a gentle introduction to quasiseparable matrices theory and does not assume any familiarity with the topic.