In 1955, John McCarthy and colleagues proposed an AI summer research project with the following aim: “An attempt will be made to find how to make machines use language, form abstractions and concepts, solve kinds of problems now reserved for humans, and improve themselves.” More than six decades later, all of these research topics remain open and actively investigated in the AI community. While AI has made dramatic progress over the last decade in areas such as vision, natural language processing, and robotics, current AI systems still almost entirely lack the ability to form humanlike concepts and abstractions—key abilities in mathematics and in everyday life. In this talk, I will argue that the inability to form conceptual abstractions—and to make abstraction-driven analogies—is a primary source of brittleness in state-of-the-art AI systems, which often struggle in adapting what they have learned to situations outside their training regimes. I will reflect on the role played by analogy-making at all levels of intelligence, and on the prospects for developing AI systems with humanlike abilities for abstraction and analogy.