TITLE: Are there prominent examples of operads in schemes?
QUESTION [18 upvotes]: There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even in $\mathbb A^{1}$-homotopy theory.  
My question is twofold: 

Are there useful examples of operads in algebraic geometry? 
Is there some intrinsic property of schemes which makes them incompatible with operatic structures? What about (higher) stacks?  

For context, my main interest is in the $E_{n}$ operads. For instance, one would naïvely imagine that the moduli space $M_{0,n}$ of genus-zero marked curves could be an algebraic geometric counterpart of the space of little disks, but there seems to be no way to define the operadic composition without passing to the Deligne–Mumford compactification of the moduli space.

REPLY [11 votes]: Mikhail Kapranov's invited lecture at the 1998 ICM was about exactly this:

Mikhail Kapranov, Operads and algebraic geometry. Documenta Mathematica Extra Volume ICM II (1998), 277-286.  (PDF of the whole volume here.)

Abstract:

The study (motivated by mathematical physics) of algebraic varieties related to the moduli spaces of curves, helped to uncover important connections with the abstract algebraic theory of operads. This interaction led to new developments in both theories, and the purpose of the talk is to discuss some of them.