This monograph explores the well-known problem of the solvability of polynomial equations. While equations up to the fourth degree are solvable, there are, as demonstrated by Niels Henrik Abel, no general algebraic formulas leading to the solution of equations of fifth or higher degree. Nevertheless, some fifth degree (quintic) equations are indeed solvable. The author describes how Galois theory can be used to identify those quintic equations that can be solved algebraically and then shows how the solutions can be found. This involves shining a light on some little known works dating back to the late 19th century, bringing new life to a classical problem. This book is a valuable resource for both students and researchers and it constitutes a good basis for a seminar on polynomials and the solvability of equations.