Publisher Summary This chapter discusses the cohomology of H-spaces, with emphasis on the ordinary cohomology with Zp = Z/pZ coefficients. It describes the consequences derived through the Hopf algebra theory. Bockstein spectral sequence is also presented. The study of the cohomology of H-spaces has several applications: (1) the fact that a space X admits an H-structure implies the existence of certain properties in its cohomology, (2) this may help to determine the cohomology of a space, knowing that it admits a multiplication, and (3) if a cohomology of a space is known, the general theory of the cohomology of H-spaces may help to determine whether or not the space admits a multiplication. The spectral sequence is constructed for generalized theories. In addition, high order cohomology operations are added to the Hopf algebra structure. The main type is that of non-stable secondary operations. In this technique, the secondary operations are replaced by primary operations in the cohomology of the projective space of the given H-space.