Algebraic topology is a young subject, and its foundations are not yet firmly in place. I shall give some history, examples and modern developments in that part of the subject called stable algebraic topology, or stable homotopy theory. This is by far the most calculationally accessible part of algebraic topology, although it is also the least intuitively grounded in visualizable geometric objects. It has a great many applications to other subjects such as algebraic geometry and geometric topology. Time will not allow me to say as much as I would like about that. Rather, I shall emphasize some foundational issues that have been central to this part of algebraic topology since the early 1960s, but that have been satisfactorily resolved only in the last few years.