Newton urged scientists never to speculate, only to prove by establishing experimental facts. Einstein urged scientists to speculate freely, since only daring speculations, not experimental facts, can advance science. Who, if either, is right? Is speculating a legitimate part of science, even in the absence of testing? If so, can speculations be evaluated without testing? How? To answer these questions, the book begins with an attempt to define the concept of speculation, a task not usually investigated by those who express strong views about speculation. The basic idea developed in the book is that speculating involves introducing assumptions, under certain “theorizing” conditions, without knowing that there is evidence for those assumptions. This idea is made precise using a concept of “evidence” that Achinstein has introduced in previous writings and also explains here. With this concept, Achinstein defends a view according to which, by contrast with Newton, speculations are crucial in science and, by contrast with Einstein, they are subject to constraints. The latter include pragmatic ones, reflecting the particular aims of the scientist in speculating, and epistemic ones that are subject to a different standard from that of “evidence sufficient for belief.” This viewpoint is illustrated and evaluated by critically examining historical and contemporary speculations in fundamental physics, as well as more general speculations within or about science, including that: nature is simple, and simplicity is a sign of truth (Newton, Einstein); a theory can only be tested “holistically” (Duhem, Quine); and there is and must be a “Theory of Everything” (string theorists, reductionists).