Abstract A finite complete rewriting system for a group is a finite presentation which gives a solution to the word problem and a regular language of normal forms for the group. In this paper it is shown that the fundamental group of an orientable closed surface of genus g has a finite complete rewriting system, using the usual generators a 1,…, a g, b 1,…, b g along with generators representing their inverses. Constructions of finite complete rewriting systems are also given for any Coxeter group G satisfying one of the following hypotheses. (1) G has three or fewer generators. (2) G does not contain a special subgroup of the form 〈 s i , s j , s k | s 2 i = s 2 j = s 2 k = ( s i s j ) 2 = ( s i s k ) m = ( s j s k ) n = 〈 with m and n both finite and not both equal to two.