In this review we discuss localization of waves by multiple scattering. In the weak case, also known as coherent backscattering, there is a twofold enhancement of the reflected intensity in the back direction. We discuss the origin of this effect, its shape with reflection angle as well as the influence of finite sample size and absorption on this shape. This discussion of the properties of the coherent backscattering cone is based on experimental investigations of multiple scattering of light in various media over the last two decades. In addition, we discuss experiments on coherent backscattering using many other types of waves. In a second part, we discuss the experimental search for strong localization in the context of past and present experiments. This leads to a presentation of the most direct method to date to show effects of strong localization via time resolved transmission measurements. These recent experiments are discussed in detail and we draw some conclusions on the nature of the localization transition. The paper ends with an outlook on possible future experiments.