In this paper the authors develop a general theory of quasi-interpolants based on trigonometric splines. Although the analysis is parallel to the familiar treatment of quasi-interpolants based on polynomial splines, the details are complicated because of the nature of trigonometric splines. In addition to developing a general theory they give an detailed treatment of two interesting classes of quasi-interpolants based on derivative formation and on simple point evaluation. The authors provide some general remarks as to how further studies in related fields can be carried on.