The purpose of this chapter will be mostly to give an idea of where logarithmic structure are useful in the theory of moduli. We will give some foundations, but only as much as needed for this purpose. It is proposed to appear in "handbook of Moduli", a book edited by Gavril Farkas and Ian Morrison. Here is an excerpt from the editors' invitation:

"The goal of the Handbook is to introduce the specialized techniques, examples and results essential to each topic, and to say enough about recent developments to prepare the reader to tackle the primary literature in the area. We are especially interested in contributions that illustrate "secret handshakes", yogas and heuristics that you use to guide intuition or simplify calculation but that are replaced by more formal arguments, or simply do not appear, in articles aimed at other specialists."

The proposed plan begins in parallel to Olsson's lectures in Pisa , and is based on our seminar :

Introduction
definitions and basic properties
differentials, smoothness, and log smooth deformations
Log smooth curves and their moduli
D semistability and log structures
Stacks of logarithmic structures
Log deformation theory in general
Rounding
Log De Rham and hodge structures
The main component of moduli spaces
log twisting and root constructions
Log stable maps