The prominent role of multiplicative cohomology theories has led to a great deal of foundational research recently on ring spectra within the field of algebraic topology. This has given rise to significant new approaches to constructing categories of spectra and ring-like objects in them. These essays contain important new contributions to the theory of structured ring spectra as well as survey papers describing relationships between them.

### Contents

1. The development of structured ring spectra Anthony Elmendorf; 2. Compromises forced by Lewis's Theorem Anthony Elmendorf; 3. Permutative categories as a model of connective stable homotopy Anthony Elmendorf and Michael Mandell; 4. Morita Theory in Abelian, derived and stable model categories Stefan Schwede; 5. Higher coherences in equivariant K-theory Michael Joachim; 6. Co-homology theories for commutative S-algebras Maria Basterra and Birgit Richter; 7. Classical obstructions and S-algebras Alan Robinson; 8. Moduli spaces of commutative ring spectra Paul Goerss and Michael Hopkins; 9. Cohomology theories for highly structured ring spectra Andrey Lazarev.