References for logarithmic geometry Hi everyone,
I'm looking for a systematical introduction to (or treatment of) logarithmic structures on schemes. I am reading Kato's article ("Logarithmic structures of Fontaine-Illusie") at the moment, but I would like to have a more detailed source that goes through or gives an overview of the constructions of classical scheme theory that have analogs in the log-setup.
Are there any articles/books that in your opinion are required reading if I want to learn about log-geometry? What are beautiful examples of applications of this machinery?
 A: One standard reference is
Luc Illusie. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic
 ́etale cohomology. Ast ́erisque, (279):271–322, 2002. Cohomologies p-adiques et applications
arithm ́etiques, II.
A: I believe that Arthur Ogus has been working on a book on this topic for many years.  I don't know if it (or at least some version of it) has appeared.  (I looked on his web-page and found what looks like a nice set of slides from a talk, 62 pages of them, but no actual book.)  
In any event, Ogus certainly has many papers on the topic.  My recommendation, if you have gotten through Kato's article, would be to start reading some of Ogus's and others' articles.
A lot of them are reasonably foundational, and should be accessible if you have Kato's article under your belt.   In addition to the names already mentioned in the various comments and answers, Kisin has a couple of nice papers using log-schemes on his web-page.
One nice application, arithmetic in nature (and the first place that I saw log schemes), is the paper of Coleman--Voloch on companion forms.  Kisin's paper on the Galois action on the
prime-to-p-etale fundamental group is another nice application to arithmetic geometry that I know of.  
A: Danny Gillam at Brown also has some nice notes on his webpage.
A: I put up some old notes by Illusie here for you; they're very detailed and treat log smoothness, the log de Rham complex, and other topics in their second exposé. They're my favourite first reference.
There is also Ogus's book, the latest draft of which is here.
