"This is a highly awaited and much appreciated fine piece of work reviewing a rapidly growing area in algebraic topology with connections to a plethora of fields."

"Overall, this article serves as an excellent introduction for anyone wishing to know more about the subject. Concepts are nicely motivated, and a wealth of references is given in case the reader wants to pursue a topic in more detail."

"Especially nice is the discussion of the chromatic behaviour of topological modular forms. Not only does the author discuss the K(1)- and K(2)-localizations of topological modular forms, but he also gives a more general treatment of stacks associated to ring spectra, following Hopkins. ... The last section of the paper reflects the growing richness of the Tmf-literature by giving a guided reading list to a large number of articles in the field. As it also contains many references to recent work, this is important added value compared to older surveys."

"This clearly written and well-structured work contains a high-level discussion of the current situation as well as active research directions in persistent homology and applied homotopy theory as seen by one of the founders of the research area."

"The chapter is a well written, useful introduction to higher categorical algebra, containing many helpful remarks about the similarity and differences of classical constructions and the new framework. No proofs are given, but results are carefully referenced. This makes it easy to get a quick overview of definitions and results before diving into the technical details of full proofs."

"As a splendid bonus, they use this to give a purely p-adic proof of Bott periodicity."

"This paper is a well-presented survey of known results of that kind, generalizing and extending Quillen's theorem. A great deal of useful background is provided, giving introductory treatments of operads, localization theory for spectra and spaces, the Bousfield-Kuhn functor, topological Andre-Quillen cohomology, Koszul duality, and partition complexes and the spectral Lie operad. This makes the paper approachable for readers who may not have much background knowledge in the subject. Some ideas are presented in ways that are insightful but not the most commonly found in the literature, making the introductory material interesting even to a reader who already has some background knowledge. Several open problems are given at the end."

"This article traverses through the concepts which are currently useful in equivariant stable homotopy theory over finite groups. This comprehensive survey may also serve as a good introduction for those interested in learning about aspects of equivariant homotopy theory"

"This is a well-written survey article about unstable motivic homotopy theory, a subject based on the idea that one can do homotopy theory in algebraic geometry, using the affine line A^1 as a substitute for the unit interval in topology."