The four basic types of auctions considered are English (ascending), Dutch (descending), and first and second price sealed bid. The basic approach is to consider auctions as games and to construct Bayesian Nash equilibriums for bidder strategies.
Menezes and Monteiro apply this to independent private values, proving revenue equivalence among other results, and then to correlated private values, with some consideration of reserve prices, entry fees, risk aversion, and the discrete valuation case (where there may be no symmetric equilibrium).
Turning to common values, they analyse examples of first- and second-price auctions, first with independent signals and then with correlated types. Independent and common values are then generalised, first to a simple model with two symmetric bidders and then to multiple bidders with "affiliated" values (a formalised notion of a global positive correlation between valuations).
A chapter on mechanism design steps up the level of abstraction by presenting a formal model of an auction rule and constructing optimal auction mechanisms for some simple cases. And a final chapter explores some of the problems raised by auctions of multiple objects.
A competence with calculus and probability theory is assumed. Appendices present proofs of some foundational results: the expected highest and second highest values for randomly distributed variables and some propositions about differential equations, affiliation, and convexity.
An Introduction to Auction Theory presents mathematical techniques and modelling approaches rather than addressing contexts or the broader picture. The examples considered are artificial constructs, not real case studies: there are only passing references to applications such as oil exploration rights (an example of common values) or government bonds (discriminatory multiple object auctions). And there's no attempt to explore applications of the theory elsewhere in economics, or in disciplines such as evolutionary biology.
The approach in An Introduction for Auction Theory works well for someone after a grasp of the mathematical foundations, however, with a few problems at the end of each chapter for use if set as a course text. It should be a good basis for tackling a broader book such as Klemperer's Auctions: Theory and Practice.