The paper is a systematic study of real hypersurfaces of quaternionic projective spaces via the focal set theory. By using the induced structures on a real hypersurface the authors obtain three classes of real hypersurfaces. Then by means of one of these classes they find an example of a proper quaternion CR-submanifold in the sense of \textit{M. Barros}, \textit{B. Y. Chen} and \textit{F. Urbano} [Kodai Math. J. 4, 399-417 (1981; Zbl 0481.53046)]. In the last part of the paper the authors obtain classification theorems for the following classes of real hypersurfaces: a) real hypersurfaces with constant principal curvatures and CR-structure in the sense of the reviewer [Chin. J. Math. 14, 81-94 (1986; Zbl 0602.53037)], b) real hypersurfaces with two distinct principal curvatures, c) pseudo-Einstein and almost Einstein real hypersurfaces.