Abstract: We provide new statistical guarantees for transfer learning via representation learning--when transfer is achieved by learning a feature representation shared across different tasks. This enables learning on new tasks using far less data than is required to learn them in isolation. Formally, we consider $t+1$ tasks parameterized by functions of the form $f_j \circ h$ in a general function class $F \circ H$, where each $f_j$ is a task-specific function in $F$ and $h$ is the shared representation in $H$. Letting $C(\cdot)$ denote the complexity measure of the function class, we show that for diverse training tasks (1) the sample complexity needed to learn the shared representation across the first $t$ training tasks scales as $C(H) + t C(F)$, despite no explicit access to a signal from the feature representation and (2) with an accurate estimate of the representation, the sample complexity needed to learn a new task scales only with $C(F)$. Our results depend upon a new general notion of task diversity--applicable to models with general tasks, features, and losses--as well as a novel chain rule for Gaussian complexities. Finally, we exhibit the utility of our general framework in several models of importance in the literature.