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This book concerns a central problem in algebraic topology-stable homotopy groups of spheres. Ever since J. P. Serre's historic result showed that the homotopy groups of spheres are finitely generated in 1950s, the determination of homotopy groups of spheres has always been the focus of algebraic topologists. Up to now, still much has not been known to us. In our book, we consider the stable part of the homotopy groups of spheres. We will mainly apply two computation tools to detect nontrivial elements of the stable homotopy groups of spheres-Adams spectral sequence and May spectral sequence. One part of this book is dedicated to obtaining some nontrivial product elements via the Adams spectral sequence. The remaining part is dedicated to approaching one important problem raised by M. Hovey--the convergence of a certain product in the Adams spectral sequence. The importance of this problem is its being related to the existence of a large seires of secondary periodic elements of the stable homotopy groups of spheres.