- Dezert-Smarandache Theory is a fusion theory, which is a natural extension of DST and works for high conflicting sources of information, and overcomes the cases where DST doesn't work. 

- Power set = is the fusion space of Dempster-Shafer Theory (DST) and Transferable Belief Model (TBM) theory; the power set is the set of all subsets of the frame of discernment, i.e. all hypotheses and all possible unions of hypotheses. {In the fusion theory union of hypotheses means uncertainty about these hypotheses.} 
- Hyper-power set = the fusion set of Dezert-Smarandache Theory (DSmT); the hyperpower set is the set formed by all unions and intersections of hypotheses. {By intersection of two or more hypotheses we understand the common part of these hypotheses - if any. In the case when their intersection is empty, we consider these hypotheses disjoint.} 
- Super-power set = the fusion space for the Unification of Fusion Theories and rules; the super-power set is the set formed by all unions, intersection, and complements of the hypotheses. {By a complement of a hypothesis we understand the opposite of that hypothesis.} 
- Basic belief assignment (bba), also called mass and noted by m(.), is a subjective probability (belief) function that a source assigns to some hypotheses or their combinations. This function is defined on the fusion space and whose values are in the interval [0, 1]. 

In the first section, we consider a frame of discernment and then we present the three known fusion spaces. The first fusion space, the power set, is used by Dempster-Shafer Theory (DST) 
and the Transferable Believe Model (TBM). The second fusion space, which is larger, the hyperpower set, is used by Dezert-Smarandache Theory (DSmT), while the third fusion space, the super-power set, is the most general one, and it is used in the Unification of Fusion Theories and Rules. In the second section we present Dempster's conditioning rule and the Bel(.) and Pl(.) functions. In order to overcome some difficult corner cases where Dempster's Conditioning Rule doesn't work, we design the first simple DSm conditioning rule and the second simple DSm conditioning rule in section 3. These rules are referring to the fact that: if a source provides us some evidence (i.e. a basic belief assignment), but later we find out that the true hypothesis is in a subset A of the fusion space, then we need to compute the conditional belief m(.|A). In section 4 we give a Class of DSm Conditioning Rules that generalizes two simple DSm conditioning rules cover. In section 5 we present two examples in military about target attribute identification. 