Integral cohomology algebras explores the algebraic objects consisting of cohomology groups with integer coefficients and the primary cohomology operations acting on them. These cohomology algebras are contrasted with their Eckmann-Hilton dual of homotopy groups and primary homotopy operations. They are also shown to generalize the Steenrod algebra structures for cohomology over finite fields. Several examples are given using the cohomology algebra structure to distinguish between topological spaces. The earlier chapters provide a comprehensive summary of the Eckmann-Hilton duality of integral cohomology and homotopy groups and the natural primary operations on them. A final chapter discusses directions for future research.