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In this monograph we use new methods to originate the definition for the tensor product of two topological semigroups so that it inherits the topological properties and we showed the existence and uniqueness of topological tensor product for two topological semigroups with identities and in general for a finite collection of topological semigroups which is somehow complicated. Then the ideal structure, minimal ideals, maximal subgroups and idempotents of the topological tensor product were explained. The concept of compactification and function spaces of a topological tensor product were investigated and in general, the relation between p-compactification of topological semigroups S and T and their tensor product are discussed, where p is an arbitrary property of compactification. Finally, by using topological tensor product technique the function spaces of Ress matrix semigroup were characterized.
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