I will discuss a recent joint work with Olivier Biquard about conic Kähler-Einstein metrics with cone angle going to zero. We study two situations, one in negative curvature (toroidal compactifications of ball quotients) and one in positive curvature (Fano manifolds endowed with a smooth anticanonical divisor) leading up to the resolution of a folklore conjecture involving the Tian-Yau metric.