I will discuss the convexity properties of momentum maps in symplectic geometry, including Atiyah - Guillemin -Sternbarg - Kirwan theorems and "nonlinear" convexity theorems, from an intrinsic point of view. The "Poisson convexity conjecture" of Weinstein will be formulated and a proof of it indicated. The local structure of (symplectic) proper groupoids plays a key role in this conjecture.