TITLE: Open questions in Riemannian geometry
QUESTION [42 upvotes]: What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.

REPLY [7 votes]: Here are two more to add to the listed above:
Conjecture of LeBrun and Salamon : 
Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?
Alekseevsky Conjecture: 
http://www2.math.ou.edu/~mjablonski/math/