Economists routinely make functional form assumptions about consumer demand to obtain welfare estimates — often for convenience, tractability, or both. How sensitive are welfare estimates to these assumptions? In this paper, we answer this question by providing bounds on welfare that hold for families of demand curves commonly considered in different literatures. We show that typical functional forms — such as linear, exponential and CES demand — are extremal in different families: they yield either the highest or lowest welfare estimate among all demand curves in those families. To illustrate the flexibility of our approach, we apply our results to the welfare analysis of trade tariffs, income taxation, and energy subsidies.