Small modular reactors (SMRs) have been proposed as a possible way to address the social problems confronting nuclear power, including poor economics, the possibility of catastrophic accidents, radioactive waste production, and linkage to nuclear weapon proliferation. Several SMR designs, with diverse technical characteristics, are being developed around the world and are promoted as addressing one or more of these problems. This paper examines the basic features of different kinds of SMRs and shows why the technical characteristics of SMRs do not allow them to solve simultaneously all four of the problems identified with nuclear power today. It shows that the leading SMR designs under development involve choices and trade-offs between desired features. Focusing on a single challenge, for example cost reduction, might make other challenges more acute. The paper then briefly discusses other cultural and political factors that contribute to the widespread enthusiasm for these reactors, despite technical and historical reasons to doubt that the promises offered by SMR technology advocates will be actually realized.