This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a {\textasciigrave }word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. It is proved that every word-hyperbolic semigroup has word problem solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.