Satisfiability (SAT) is the canonical NP-Complete problem. Many other interesting and practical problems can be encoded into SAT. This is why it is important from a theoretical and practical point of view. Nowadays there are very good algorithms to solve SAT, the best from the point of view of industrial instances can be shown equivalent to the proof system Resolution. We will present the algorithm, and the equivalence to Resolution. Other proof systems are more powerful than Resolution, and still might be amenable to finding refutation algorithms, like Cutting Planes. We will talk about posible ways to automatize Cutting Planes. Also we will present the dual-rail encoding of formulas, to be then refuted using MaxSAT algorithms, as another methodology for SAT more powerful than Resolution.