We present a theoretical study of the electronic and excitonic states in InAs/GaAsSb quantum dots. We first center our study on the dependence of the antimony composition in the positioning of conduction- and valence-band alignments in InAs/GaAsSb/GaAs heterostructures. We predict a transition from type I to type II quantum dots at critical composition xc=0.128, which describes well the experimental trend. We discuss the influence of the quantum dot size and antimony composition on the spatial distributions of carriers and the exciton binding energy. We find that the ground state exciton binding energy is always significantly smaller ≃4meV for type II than for corresponding type I quantum dots ≃14meV. Finally, we also predict the excitonic radiative lifetime and find 1 ns for type I and 10 ns for type II quantum dots, in agreement with the existing experimental literature.