This is an expository paper about secondary cohomology operations and shows a little of how they can be used to prove theorems about the cohomology of H-spaces at the prime 2. A second theme is to show how results involving cohomology operations can be established using the theory of H-spaces. The paper is recommended to any graduate student or established topologist who wishes to gain insight into an area of considerable complexity but which has produced some excellent results in recent years particularly in papers of the author, Lin and Kane.