TITLE: Why does so much recent work involve K3 surfaces?
QUESTION [38 upvotes]: I've been noticing that a whole lot of papers published to the Arxiv recently involve K3 surfaces. Can anyone give me (someone who, at this point, knows little more about K3 surfaces than their definition) an idea why they are coming up so often?  
Some questions that might be relevant: Are there particular reasons that they are so important?  Are there special techniques that are available for K3 surfaces, but not more generally, making them easier to study? Are they just "in vogue" at the moment? Are they more like a subject of research (e.g., people are carrying out some sort of program to better understand K3 surfaces) or a testing ground (people with ideas in all sorts of different areas end up working the ideas out over K3 surfaces, because more general versions are much more difficult)?

REPLY [11 votes]: K3 surfaces are also interesting from the point of view of complex dynamics. To quote from Curtis T. McMullen's  introduction  to his paper  ``Dynamics on K3 surfaces: Salem numbers and Siegel disks", Journal fur die Reine und Angewandte Mathematik 2002(545): 201–233, 
"The first dynamically interesting automorphisms of compact
complex manifolds arise on K3 surfaces.
Indeed, automorphisms of curves are linear (genus 0 or 1) or of finite
order (genus 2 or more). Similarly, automorphisms of most surfaces (including
$\mathbb{P}^2$, surfaces of general type and ruled surfaces) are either linear, finite
order or skew-products over automorphisms of curves. Only K3 surfaces,
Enriques surfaces, complex tori and certain non-minimal rational surfaces
admit automorphisms of positive topological entropy [Ca2]. The automorphisms of tori are linear, and the Enriques examples are double-covered by
K3 examples."
In this paper McMullen gives examples of K3 surfaces admitting automorphisms with Siegel disks (i.e., domains on which the automorphism is conjugate to a rotation). There are countably many such surfaces, all of them non-projective. The citation [Ca2] is to the paper  by S. Cantat, Dynamique des automorphismes des surfaces
projectives
complexes.
CRAS Paris S ́er. I Math.
328
(1999), 901–906, which grew into a larger work: S. Cantat : Dynamique des automorphismes des surfaces K3 ; Acta Math. 187:1 (2001), 1--57
These papers are only a few examples (perhaps of landmark character);  there  has been a lot of study of dynamics on K3 surfaces going on  indeed.

REPLY [5 votes]: In the mathematical physics community, there was recently a resurgence of interest for K3-surfaces due to the observation of Egushi, Ooguri and Tashikawa that the elliptic genus of K3-surfaces seems to be built out of representations of the Mathieu group M24. This phenomenon has been dubbed the "Mathieu moonshine". There is still no definitive understanding of this relation, as far as I know.