For integers k, g, d, a (k; g, d)-cage (or simply girth-diameter cage) is a smallest k-regular graph of girth g and diameter d (if it exists). The order of a (k; g, d)-cage is denoted by n(k; g, d). We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed k and g. We theoretically determine the exact values n(3; g, d), and count the number of corresponding girth-diameter cages, for \(g \in \{4,5\}\) . Moreover, we design and implement an exhaustive graph generation algorithm and use it to determine the exact order of several open cases and obtain – often exhaustive – sets of the corresponding girth-diameter cages. The largest case we generated and settled with our algorithm is a (3; 7, 35)-cage of order 136.