In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if (X,*) is a groupoid and if varphi : X^2rightarrow X^2 is a function varphi (a, b) = (u, v), then (X,*) is a left-twisted semigroup with respect to varphi if for all a, b, cin X, a*(b*c) = (u*v)*c. Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule (a*b)*c = u*(v*c) for all a, b, cin X. Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest is the class of groupoids defined over a field (X,+, cdot ) via a formula x*y=lambda x + mu y, with lambda , mu in X, fixed structure constants. Properties of these groupoids as twisted semigroups are discussed with several results of interest obtained, e.g., that in this setting simultaneous left-twistedness and right-twistedness of (X,*) implies the fact that (X,*) is a semigroup.