Ultrasonic experiments are well suited to the investigation of classical wave transport through strongly scattering media, and are playing a role that is often complementary to investigations using light or microwaves. Advantages of ultrasonic techniques are their ability to readily detect the wave field (not just the intensity), to perform experiments resolved in both time and space, and to control the properties of the medium being investigated over a wide range of scattering contrasts. This first paper reviews what has been learned from ultrasonic experiments over the last 15 years about the ballistic and diffusive propagation of classical waves through strongly scattering disordered media. These results are compared with studies of ordered media (phononic crystals), where band gaps and super-resolution focusing have been observed.