With user-facing apps adopting serverless computing, good latency performance of serverless platforms has become a strong fundamental requirement. However, it is difficult to achieve this on platforms today due to the design of their underlying control and data planes that are particularly ill-suited to short-lived functions with unpredictable arrival patterns. We present Atoll, a serverless platform, that overcomes the challenges via a ground-up redesign of the control and data planes. In Atoll, each app is associated with a latency deadline. Atoll achieves its per-app request latency goals by: (a) partitioning the cluster into (semi-global scheduler, worker pool) pairs, (b) performing deadline-aware scheduling and proactive sandbox allocation, and (c) using a load balancing layer to do sandbox-aware routing, and automatically scale the semi-global schedulers per app. Our results show that Atoll reduces missed deadlines by ~66x and tail latencies by ~3x compared to state-of-the-art alternatives.