Rigidity theory is the study of the uniqueness of structures. In most cases, a structure is defined by a set of variables with constraints, ideally polynomial, that come from geometry. A constraint can be an equality or an inequality. There are various types of rigidity. Intuitively, local rigidity means uniqueness in a small neighborhood, global rigidity means uniqueness in a much larger space at the given dimension, and universal rigidity means uniqueness in all higher dimensions. This thesis explores the rigidity of several common structures that are slightly more complex than a set of points with fixed distance constraints, known as the bar-joint frameworks. These structures include circle packings, polyhedra, and various special sets of points with inequality constraints, known as tensegrities. They are natural extensions of the bar-joint frameworks that occur in many studies. The rigidity of circles with a given tangency pattern and fixed radii has been well studied. This is known as sticky disks, as the disks that are required to be tangent must stick together''. It is natural to ask if the rigidity results for sticky disks still hold with flexible radii. Another problem arises from the study of polytopes as regards the rigidity of a polyhedron with fixed edge lengths and vertices of each facet staying in the same plane. To solve these problems, we extend the methods used for bar-joint frameworks so that the algebra behaves analogously. Several examples of structures with interesting results on rigidity are given in each chapter. Often, rigidity is done in a generic'' sense where singularities are ignored. An ambitious attempt is made to replace the assumption generic'' with the assumption convex'' for several classes of bar-joint frameworks. Some examples of resolved cases and an open case are given in the last chapter.