equivalent to theory B compactified on a space of small volume (and vice versa). This means, for example, that some other scalar field $t$, the exponential of whose vev determines the volume of the compactified dimensions, satisfies $t_A =  - t_B$. $T$ dualities can be checked order-by-order in string perturbation theory, and therefore they were the first ones to be understood. Theories A and B can be called $U$ dual if theory A compactified on a space of large (or small) volume is equivalent to theory B at strong (or weak) coupling. In this case $t_A = \pm \phi_B$. This is not exactly the definition of $U$ duality that was originally proposed, but I feel it is in the spirit of the original proposal and find it to be convenient. When present, each of these dualities is supposed to constitute an exact quantum equivalence, which means that the two ‘theories’ should really be viewed as different descriptions of a single theory. It sometimes happens that a single theory is self-dual under a group of these dualities. In this case, the dualities are symmetries — discrete gauge symmetries, to be precise. This means that configurations related by duality transformations describe equivalent 