We consider the following augmentation problem: Given a rigid graph $G=(V,E)$, find a minimum cardinality edge set $F$ such that the graph $G'=(V,E\cup F)$ is globally rigid. We provide a min-max theorem and a polynomial-time algorithm for this problem for several types of rigidity, such as rigidity in the plane or on the cylinder. Rigidity is often characterized by some sparsity properties of the underlying graph, and global rigidity is characterized by redundant rigidity (where the graph remains rigid after deleting an arbitrary edge) and 2- or 3-vertex-connectivity. Hence, to solve the above-mentioned problem, we define and solve polynomially a combinatorial optimization problem family based on these sparsity and connectivity properties. This family also includes the problem of augmenting a $k$-tree-connected graph to a highly $k$-tree-connected and 2-connected graph. Moreover, as an interesting consequence, we give an optimal solution to the so-called global rigidity pinning problem, where we aim to find a minimum cardinality vertex set $X$ for a rigid graph $G=(V,E)$, such that the graph $G+K_X$ is globally rigid in $\mathbb{R}^2$ where $K_X$ denotes the complete graph on the vertex set $X$.