The hardware of computers, e.g. circuits, sequential circuits or VLSI chips, realizes Boolean functions. The design of efficient hardware is a fundamental issue in computer design. Because of the large cost for the physical construction of a new chip, the logical synthesis and the verification as well as the generation of test patterns have to be performed before the chip is built. For these purposes data structures for Boolean functions supporting operations like the evaluation on a given input, the satisfiability test, the synthesis of a representation for f = g h for some binary operator from representations for g and h are necessary tools. The corresponding state of the art data structure is the ordered binary decision diagram (OBDD). Efficient algorithms for the operations on OBDDs and the expressive power of OBDDs of polynomial size are discussed. A generalized data structure called graph-driven binary decision diagram is presented. The new data structure allows for many important functions a representation of polynomial size, even for functions whose representations by OBDDs have exponential size. Efficient algorithms for the operations on graph driven BDDs are outlined.