Contemporary satisfiability solvers are the corner-stone of many successful applications in domains such as automated verification and artificial intelligence. The impressive advances of SAT solvers, achieved by clever engineering and sophisticated algorithms, enable us to tackle Boolean Satisfiability (SAT) problem instances with millions of variables – which was previously conceived as a hopeless problem. We provide an introduction to contemporary SAT-solving algorithms, covering the fundamental techniques that made this revolution possible. Further, we present a number of extensions of the SAT problem, such as the enumeration of all satisfying assignments (ALL-SAT) and determining the maximum number of clauses that can be satisfied by an assignment (MAX-SAT). We demonstrate how SAT solvers can be leveraged to solve these problems. We conclude the chapter with an overview of applications of SAT solvers and their extensions in automated verification.