In the appendix, we give a short account of model categories, summarising the definitions and results that we use in this book.

In this chapter, we introduce symmetric spectra and orthogonal spectra along with their associated stable model structures. These versions of spectra have various technical advantages over sequential spectra. Furthermore, they are Quillen equivalent to the category of sequential spectra (equipped with its stable model structure). Hence, one may choose between these models according to their relative strengths. The primary advantage of symmetric and orthogonal spectra is that these model categories are symmetric monoidal models for the stable homotopy category. We will examine these monoidal structures further later on and show that symmetric spectra and orthogonal spectra are monoidally Quillen equivalent. Several other models of spectra also exist, and we will give short introductions to these later in this chapter. We end the chapter with a result that, roughly speaking, says that any model for the stable homotopy category will be Quillen equivalent to sequential spectra.