A three-link, single-degree-of-freedom mechanism, named the Spirograph linkage, is investigated in this thesis. What is special about this open-chain mechanism is that, by the use of a chain and two sprockets and therefore maintaining a constant angular velocity ratio between the two rotating links, the path traced by this mechanism can be shown to be either an epitrochoid or a hypotrochoid. Through the understanding of the formulations of the trochoids, the relations between the design parameters of the Spirograph linkage and the corresponding path patterns are derived. With certain design parameter combinations, the Spirograph linkage is able to feature paths with cusps, which means dwells and are useful in applications. A design chart for such cusp-generating Spirograph linkages is then included in the thesis for the sake of convenience. In addition, by matching the link-length ratio with the angular velocity ratio, ways for finding closed single-loop paths of the Spirograph linkage are also studied. As for the path generation problems concerning only a fraction of the entire path, we compare results given by the classical precision point scheme with assorted combinations of initial conditions. In the latter part of the thesis, non-circular sprockets are also introduced to provide the Spirograph linkage additional freedom to cope with the demand of more flexible, i.e., non-symmetrical shapes of the path. However, many constraints are to be imposed on the design process of such linkages, and these design limitations are elaborated in the thesis. At last, the use of Newton¡¦s method, the selection of the proper type of trochoids, the application of the envelop method for obtaining the profile of the non-circular sprocket, the analyses of angular velocity ratios, etc. are all exemplified in the several numerical examples of the thesis.