Amazon Aurora Serverless is an on-demand, autoscaling configuration for Amazon Aurora with full MySQL and PostgreSQL compatibility. It automatically offers capacity scale-up/down (i.e., vertical scaling) based on a customer database application's needs. For customers with time-varying workloads, it offers cost savings compared to provisioned Aurora or other alternatives due to its agile and granular scaling and its usage-based charging model. This paper describes the key ideas underlying Aurora Serverless's resource management. To help meet its goals, Aurora Serverless adapts and fine tunes well-established ideas related to resource over-subscription; reactive control informed by recent measurements; distributed & hierarchical decision-making; and innovations in the DB engine, OS, and hypervisor for efficiency. Perhaps the most challenging goal is to offer a consistent resource elasticity experience while operating hosts at high degrees of utilization. Aurora Serverless implements several novel ideas for striking a balance between these opposing needs. Its technique for mapping workloads to hosts ensures that, in the common case, there is adequate spare capacity within a host to support fast scale-up for a workload. In the rare event this is not so, it live migrates workloads to ensure seamless scale-up. Its load distribution strategy is characterized by "unbalancing" of load across hosts to enable agile live migrations. Finally, it employs a token bucket-based rate regulation mechanism to prevent a growing workload from saturating its host faster than live migration-based remedial actions.