The evaporation of a sessile droplet of liquid is a complex and multifaceted fundamental topic of enduring scientific interest that is key to numerous physical and biological processes. As a result, in recent decades a considerable multidisciplinary research effort has been directed toward many different aspects of the problem. This review focuses on some of the insights that can be obtained from relatively simple mathematical models and discusses some of the directions in which the field may move in the future. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 55 is January 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Motivated by the industrial manufacture of organic light-emitting-diode displays, we formulate and analyse a mathematical model for the evolution of a thin droplet in a shallow axisymmetric well of rather general shape both before and after touchdown that accounts for the spatially non-uniform evaporation of the fluid, perform physical experiments using three cylindrical wells with different small aspect ratios, and validate the mathematical model by comparing the present experimental results with the corresponding theoretical predictions for a cylindrical well.