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The theory of hypergroups is a rapidly developing area of mathematics due to its diverse applications in different areas like probability, harmonic analysis, etc. This book exhibits the use of functional equations and spectral synthesis in the theory of hypergroups. It also presents the fruitful consequences of this delicate “marriage” where the methods of spectral analysis and synthesis can provide an efficient tool in characterization problems of function classes on hypergroups. This book is written for the interested reader who has open eyes for both functional equations and hypergroups, and who dares to enter a new world of ideas, a new world of methods — and, sometimes, a new world of unexpected difficulties. Table of Contents Introduction Polynomial Hypergroups in One Variable Polynomial Hypergroups in Several Variables Sturm-Liouville Hypergroups Two-Point Support Hypergroups Spectral Analysis and Synthesis on Polynomial Hypergroups Spectral Analysis and Synthesis on Sturm-Liouville Hypergroups Moment Problems on Hypergroups Special Functional Equations on Hypergroups Difference Equations on Polynomial Hypergroups Stability Problems on Hypergroups