Modern language models have demonstrated remarkable reasoning capabilities by using chain-of-thought (CoT). One hypothesis about the inner workings of CoT is that it breaks down originally complex tasks into smaller subtasks that are more amenable to learning. We formalize this notion by showing possibility and impossibility results of learning from in-context demonstrations with and without CoT. In particular, with CoT, we examine a family of learning algorithms that learn a task step-by-step, capable of composing simpler functions from individual reasoning steps to form an overall complex function. This process reduces the difficulty of learning a task to that of the hardest reasoning step in the chain. Moreover, we prove Transformers can express this algorithm and thus they can efficiently in-context learn arbitrary tasks as long as these tasks can be decomposed into a finite number of subtasks, each of which are efficiently learnable. In contrast, without CoT, we demonstrate that there exist tasks that are inherently unlearnable by the same algorithm. Overall, our results suggest several provably effective ways for decomposing target problems to instantiate CoT. Empirically, we demonstrate our proposed CoT construction significantly enhances the reasoning capabilities of real-world LLMs in solving challenging arithmetic reasoning tasks, including learning polynomials and Boolean formulas.