We will discuss three properties: amenability, property A (exactness) and coarse embeddability into a Hilbert space. Property A is a non-equivariant version of amenability and is satisfied by a large class of groups and metric spaces. Coarse embeddability follows from property A and it was an open question for some time whether they are equivalent. We will present the progress in finding examples of spaces without property A but coarsely embeddable. This includes my recent examples of warped cones with such properties. Warped cones, which will be carefully introduced, are metric spaces emerging from a group action via gluing edges between points in each orbit.