This paper deals with the decidability of semigroup freeness. More precisely, thefreeness problem over a semigroup S is defined as: given a finite subsetX ⊆ S, decide whether each element ofS has at most one factorization over X. To date, thedecidabilities of the following two freeness problems have been closely examined. In 1953,Sardinas and Patterson proposed a now famous algorithm for the freeness problem over thefree monoids. In 1991, Klarner, Birget and Satterfield proved the undecidability of thefreeness problem over three-by-three integer matrices. Both results led to the publicationof many subsequent papers. The aim of the present paper is (i) to presentgeneral results about freeness problems, (ii) to study the decidabilityof freeness problems over various particular semigroups (special attention is devoted tomultiplicative matrix semigroups), and (iii) to propose precise,challenging open questions in order to promote the study of the topic.