that modern technology makes it possible to develop an analogous theory of orientations for ${A_{\infty}}$ ring spectra. We also discovered that recent work on $\infty$-categories provides an excellent framework for the study of Thom spectra and orientations, allowing constructions which are very close to the original sketch in Sullivan’s notes.

In this paper we review and extend the theory of orientations of [@MQRT:ersers], taking advantage of intervening technical developments, particularly in the theory of multiplicative spectra and of $\infty$-categories. We recover the obstruction theory of [@MQRT:ersers], expressed in terms of the adjunction of Theorem \[t-th-units-intro\], and extend it to ${A_{\infty}}$ ring spectra. We also extend the theory to more general kinds of fibrations, such as those which appear in the study of twisted generalized cohomology.

We begin by reviewing the analogy between the theory of orientations and the theory of locally free sheaves of rank one. This analogy, which appears in Sullivan’s notes, is an excellent guide to the basic results of the subject. Modern technology enables us to hew more closely to this intuitive picture than was possible thirty years ago, and so it is remarkable how much of the theory was worked out in the early seventies.

The 