Boolean-width is similar to clique-width, rank-width and NLC-width in that all these graph parameters are constantly bounded on the same classes of graphs. In many classes where these parameters are not constantly bounded, boolean-width is distinguished by its much lower value, such as in permutation graphs and interval graphs where boolean-width was shown to be O(log n) . Together with FPT algorithms having runtime O*(cboolw) for a constant c this helped explain why a variety of problems could be solved in polynomial-time on these graph classes. In this paper we continue this line of research and establish non-trivial upper-bounds on the boolean-width and linear boolean-width of any graph. Again we combine these bounds with FPT algorithms having runtime O*(cboolw), now to give a common framework of moderately-exponential exact algorithms that beat brute-force search for several independence and domination-type problems, on general graphs. Boolean-width is closely related to the number of maximal independent sets in bipartite graphs. Our main result breaking the triviality bound of n/3 for boolean-width and n/2 for linear boolean-width is proved by new techniques for bounding the number of maximal independent sets in bipartite graphs.