  These notes give an introduction to embedded contact homology (ECH) of
contact three-manifolds, gathering together many basic notions which are
scattered across a number of papers. We also discuss the origins of ECH,
including various remarks and examples which have not been previously
published. Finally, we review the recent application to four-dimensional
symplectic embedding problems. This article is based on lectures given in
Budapest and Munich in the summer of 2012, a series of accompanying blog
postings at floerhomology.wordpress.com, and related lectures at UC Berkeley in
Fall 2012. There is already a brief introduction to ECH in the 2010 ICM
proceedings, but the present notes give much more background and detail.
