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ii iii Abstract. Let S be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category M S of "S-modules" whose derived category D S is equivalent to the classical stable homotopy category. This allows a simple and algebraically manageable definition of "S-algebras" and "commutative S-algebras" in terms of associative, or associative and commutative, products R ∧ S R −→ R. These notions are essentially equivalent to the earlier notions of A ∞ and E ∞ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of R-modules in terms of maps R ∧ S M −→ M .When R is commutative, the category M R of R-modules also has an associative, commutative, and unital smash product, and its derived category D R has properties just like the stable homotopy category.Working in the derived category D R , we construct spectral sequences that specialize to give generalized universal coefficient and Künneth spectral sequences. Classical torsion products and Ext groups are obtained by specializing our constructions to Eilenberg-Mac Lane spectra and passing to homotopy groups, and the derived category of a discrete ring R is equivalent to the derived category of its associated Eilenberg-Mac Lane S-algebra.We also develop a homotopical theory of R-ring spectra in D R , analogous to the classical theory of ring spectra in the stable homotopy category, and we use it to give new constructions as MU-ring spectra of a host of fundamentally important spectra whose earlier constructions were both more difficult and less precise.Working in the module category M R , we show that the category of finite cell modules over an S-algebra R gives rise to an associated algebraic K-theory spectrum KR. Specialized to the Eilenberg-Mac Lane spectra of discrete rings, this recovers Quillen's algebraic K-theory of rings. Specialized to suspension spectra Σ ∞ (ΩX) + of loop spaces, it recovers Waldhausen's algebraic K-theory of spaces.Replacing our ground ring S by a commutative S-algebra R, we define Ralgebras and commutative R-algebras in terms of maps A ∧ R A −→ A, and we show that the categories of R-modules, R-algebras, and commutative R-algebras are all topological model categories. We use the model structures to study Bousfield localizations of R-modules and R-algebras. In particular, we prove that KO and KU are commutative ko and ku-algebras and therefore commutative S-algebras.We define the topological Hochschild homology R-module T HH R (A; M ) of A with coefficients in an (A, A)-bimodule M and give spectral sequences for the calculation of its homotopy and homology groups. Again, classical Hochschild homology and cohomology groups are obtained by specializing the constructions to Eilenberg-Mac Lane spectra and passing to homotopy groups. iv

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Working in the category T of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functorsD→T for a suitable small topological categoryD. WhenD is symmetric monoidal, there is a smash product that gives the category of D‐spaces a symmetric monoidal structure. Examples include prespectra, as defined classically, symmetric spectra, as defined by Jeff Smith, orthogonal spectra, a coordinate‐free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category, Γ‐spaces, as defined by Graeme Segal, W‐spaces, an analogue of Γ‐spaces with finite sets replaced by finite CW complexes in the domain category. We construct and compare model structures on these categories. With the caveat that Γ‐spaces are always connective, these categories, and their simplicial analogues, are Quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category. Monoids in these categories are (strict) ring spectra. Often the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. When this holds, the respective categories of ring and module spectra are Quillen equivalent and thus have equivalent homotopy categories. This allows interchangeable use of these categories in applications. 2000Mathematics Subject Classification: primary 55P42; secondary 18A25, 18E30, 55U35.

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