A new formalism for encoding quantum theories in the general boundary formulation (GBF) is developed. It is analogous to the mixed state formalism of the standard formulation. The positivity of the probabilities is directly expressed in terms of order structures on vector spaces. This explain the name of positive formalism in contrast with the usual amplitude formalism. The positive formalism is introduced first for purely bosonic quantum theory which is considered simpler and conceptually more transparent than the general case. Then, the theoretical framework is extended to include the fermionic degrees of freedom. In both cases, the probability interpretation is discussed for two new basic objects: the probability map and the expectation map. The projection operators play a more fundamental role in the probability interpretation of the GBF than the states. It is emphasized that the GBF is motivated by the problem of providing a suitable foundation for a quantum theory of gravity. Also, it is suggested that the new formalism opens the way for introducing the GBF into the quantum theory of information.