  Hadamard spaces have traditionally played important roles in geometry and
geometric group theory. More recently, they have additionally turned out to be
a suitable framework for convex analysis, optimization and nonlinear
probability theory. The attractiveness of these emerging subject fields stems,
inter alia, from the fact that some of the new results have already found their
applications both in mathematics and outside. Most remarkably, a gradient flow
theorem in Hadamard spaces was used to attack a conjecture of Donaldson in
Kahler geometry. Other areas of applications include metric geometry and
minimization of submodular functions on modular lattices. There have been also
applications into computational phylogenetics and imaging.
  We survey recent developments in Hadamard space analysis and optimization
with the intention to advertise various open problems in the area. We also
point out several fallacies in the existing proofs.
