Many techniques for verifying properties of continuous-time systems are limited to systems of moderate size. In this paper, we propose an approach based on assume-guarantee contracts and compositional reasoning for verifying properties of a broad class of continuous-time systems consisting of interconnected components. The notion of assume-guarantee contracts makes it possible to divide responsibilities among the system components: a contract specifies the property that a component must fulfill under some assumptions on the behavior of its environment (i.e. of the other components). We define weak and strong semantics of assume-guarantee contracts. We then establish a certain number of results for compositional reasoning, which allow us to show that a global assume-guarantee contract of the whole system is satisfied when all components satisfy their own contracts. We show that the weak satisfaction of the contract is sufficient to deal with interconnections described by a directed acyclic graph, while strong satisfaction is needed to reason about general interconnections containing cycles. Specific results for systems described by differential inclusions and invariance assume-guarantee contracts are then developed. Finally, we show how the proposed assume-guarantee framework can recast different versions of the small-gain theorem as a particular case. Throughout the paper, the main results are illustrated using simple examples.