Given a discrete group action on a compact metric space, John Roe constructs an unbounded space by rescaling the metric and "warping" it by adding shortcuts between points in an orbit. Geometry of the obtained space, called the warped cone, reflects properties of the action and the group -- amenability of the action results in property A, while spectral gap leads to coarse non-embeddability into Banach spaces. A well-studied geometric object capturing group properties is its box space (sequence of finite quotients). Warped cones generalise them in a sense, as for an appropriate profinite action, the box space embeds quasi-isometrically into the warped cone. This yields some interesting examples.