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The class (resp., t-class) semigroup of an integral domain is the semigroup of the isomorphy classes of the nonzero fractional ideals (resp., t-ideals) with the operation induced by ideal (t-) multiplication. This paper surveys recent literature which studies ring-theoretic conditions that reflect reciprocally in the Clifford property of the class (resp., t-class) semigroup. Precisely, it examines integral domains with Clifford class (resp., t-class) semigroup and describes their idempotent elements and the structure of their associated constituent groups. [ABSTRACT FROM AUTHOR]