When studying the propositional satisfiability problem (SAT), that is, the problem of deciding whether a propositional formula is satisfiable, it is typically assumed that the formula is given in the conjunctive normal form (CNF). Also most software tools for deciding satisfiability of a formula (SAT solvers) assume that their input is in CNF. An important reason for this is that it is simpler to develop efficient data structures and algorithms for CNF than for arbitrary formulas. On the other hand, using CNF makes efficient modeling of an application cumbersome. Therefore one often employs a more general formula representation in modeling and then transforms the formula into CNF for SAT solvers. Transforming a propositional formula in CNF either increases the formula size exponentially or requires the use of auxiliary variables, which can have an negative effect on the performance of a SAT solver in the worst-case. Moreover, by translating to CNF one often loses information about the structure of the original problem. In this chapter we survey methods for solving propositional satisfiability problems when the input formula is not given in CNF but as a general formula or even more compactly as a Boolean circuit. We show how the techniques applied in CNF level Davis-Putnam-Loveland-Logemann algorithm generalize to Boolean circuits and how the problem structure available in the circuit form can be exploited. Then we consider a closely related area of automatic test pattern generation (ATPG) for digital circuits and review classical ATPG algorithms, formulation of ATPG as a SAT problem, and advanced techniques for SAT-based ATPG.