We discuss the results of recent studies of acoustic turbulence in a system of nonlinear second sound waves in a high-quality resonator filled with superfluid 4He. It was found that, when the driving amplitude was sufficiently increased, a steady-state direct wave cascade is formed involving a flux of energy towards high frequencies. The wave amplitude distribution follows a power law over a wide range of frequencies. Development of a decay instability at high driving amplitudes results in the formation of subharmonics of the driving frequency, and to a backflow of energy towards the low-frequency spectral domain, in addition to the direct cascade.