Abstract: Suppose you have a structure made of rigid bars joined at fully-rotational joints. The obvious basic question is whether the structure is rigid or flexible. Mathematically, this question translates to: given a collection of points \(p_1, \dots, p_n \in \mathbb{R}^d\) and some choice of pairs of points, can you continuously move the \(p_i\) while preserving the distances between the chosen pairs of points without the motion being trivial, i.e. coming from a rigid motion (Euclidean isometry) of \(\mathbb{R}^d\)? For an arbitrary such structure (a “framework”), this can be surprisingly difficult, but if the \(p_i\) are “generic”, then it only depends on the underlying graph, and there are nice conditions and algorithms to determine if the framework is rigid for e.g. \(d = 2\). I'll discuss some of the known results about the rigidity of frameworks, some open problems, and weird behaviors in higher dimensions.