Previous research into bounded suboptimal search has focused on the development of epsilon-admissible algorithms which are guaranteed to return solutions that are no more than a factor larger than optimal. In this paper, we consider the problem of how to construct search algorithms that satisfy alternative types of guarantees such as an additive bound. This bounding paradigm requires that the cost of any solution found is no more than the optimal cost plus gamma, which is a user-defined constant. To this end, we provide theorems that define sufficient conditions for developing algorithms for arbitrary bounding paradigms when using best-first search, iterative deepening, or focal list-based search. We then show by experimentation that these theorems can be used to construct effective additively bounded algorithms.