I discuss (the full range of) Caffarelli, Kohn, Nirenberg's inequalities for fractional Sobolev spaces and improvements of Hardy's and Caffarelli, Kohn, Nirenberg's inequalites for (standard integer order) Sobolev spaces with proofs. Interestingly, the proofs are quite elementary and mainly based on the Poincaré and Sobolev inequalities for an annulus; the integration-by-part arguments are not required. This is joint work with Marco Squassina.