A detailed presentation of the Landau theory of the cholesteric blue phase (BP) is given with emphasis on the comparison of model predictions with reported experimental data. The basic model, which is asymptotically exact in the high-chirality limit, leads to the prediction of a body-centered cubic (bcc) $O^5 \mathrm{}\left(I432\mathrm{}\right)\mathrm{}$ structure for the BP. However, the model can be systematically extended to physical chiralities and used to calculate the free energies of more complex cubic structures. In particular, bcc $O^8 \left(I{4}_{1}32\right)$ and simple-cubic $O^2 \left(P{4}_{2}32\right)$ structures are analyzed and their optical and thermodynamic properties, as well as those of $O^5$, are compared with those reported for the two observed phases, BPI and BPII. It is shown that, in general, the known properties of these phases are consistent with a structure assignment of $O^8$ for BPI and $O^2$ for BPII. Some promising lines for future investigation are indicated.