TITLE: Why is the Laplacian ubiquitous?
QUESTION [35 upvotes]: The title says it all. 
I'm wondering why the Laplacian appears everywhere, e.g. number theory, Riemannian geometry, quantum mechanics, and representation theory. And people seems to care about their eigenvalues. Is there any deep reason behind this?

REPLY [3 votes]: The Laplacian is a second order differential operator which is both linear
and invariant under any rigid body motion (rotations and translations). 
There is no simpler non-trivial operator with these properties. In modeling many
physical phenomena, one expects these two properties to arise, so sometimes
the simplest approximation based on physical principles leads to the Laplacian.