Economists routinely make functional form assumptions about consumer demand to obtain welfare estimates. How sensitive are welfare estimates to these assumptions? We answer this question by providing bounds on welfare that hold for families of demand curves commonly considered in different literatures. We show that commonly chosen 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 our approach, we apply our results to the welfare analysis of energy subsidies, trade tariffs, pensions, and income taxation.