TITLE: What are the big problems in probability theory?
QUESTION [108 upvotes]: Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the Poincaré conjecture for a long time, and currently has the classification of 4-manifolds. PDE theory has the Navier-Stokes equation to deal with.
So what are the big problems in probability theory and stochastic analysis? 
I'm a grad student working in the field, but I can't name any major unsolved conjectures or open problems which are driving research. I've heard that stochastic Löwner evolutions are a big field of study these days, but I don't know what the conjectures or problems relating to them are. 
Does anyone have any suggestions?

REPLY [4 votes]: Forming tools for handling random surfaces and proving an universal central limit theorem for them given minimal conditions (think usual CLT):
a)The Gaussian Free Field (GFF) has shown up as the limiting universal object for many random surfaces (in KPZ 2+1, random tilings, random matrix theory Ginibre ensembles cf Borodin's and Kenyon's work)
b)Schramm-Loewner evolutions (SLE) have shown up as the limiting interfaces for families of statistical models.
c)Finally, merging the above two pictures since SLEs can be coupled to GFF (cf Sheffield).
Another powerful result would be showing the equivalence of Random planar maps and Liouville quantum gravity (LQG) (a promising approach by Miller and Sheffield). This is because it happens that statistical models become easier to handle over these random surfaces (Kazanov-Ising model, LERW-Duplantier).