In the intersection of mathematical logic and computer science, this book investigates the search problems and reducibilities among them that have known or potential relevance to bounded arithmetic theories. The same structures are viewed from two very different angles: that of computational complexity, and that of sets of low complexity consequences of weak logical theories, bounded arithmetics. Two distinct techniques of characterization of such sets by search problems are presented, with Herbrand's theorem at the root of both. Additional attention is paid to search problems from the minimization family, although their logical counterparts are mostly still to be discovered. In this way, the two worlds throw light onto each other.