We review studies of the generation and propagation of nonlinear and shock sound waves in He II (the superfluid phase of 4He), both under the saturated vapor pressure (SVP) and at elevated pressures. The evolution in shape of second and first sound waves excited by a pulsed heater has been investigated for increasing power W of the heat pulse. It has been found that, by increasing the pressure P from SVP up to 25 atm, the temperature T_lambda, at which the nonlinearity coefficient alpha of second sound reverses its sign, is decreased from 1.88 to 1.58 K. Thus at all pressures there exists a wide temperature range below T_lambda where alpha is negative, so that the temperature discontinuity(shock front) should be formed at the center of a propagating bipolar pulse of second sound. Numerical estimates show that, with rising pressure, the amplitude ratio of linear first and second sound waves generated by the heater at small W should increase significantly. This effect has allowed us to observe at P=13.3 atm a linear wave of heatingrarefaction in first sound, and its transformation to a shock wave of cooling (compression). Measurements made at high W for pressures above and below the critical pressure in He II, Pcr=2.2 atm, suggest that the main reason for initiation of the first sound compression wave is strong thermal expansion of a layer of He I (the normal phase) created at the heater-He II interface when W exceeds a critical value. Experiments with nonlinear second sound waves in a high-quality resonator show that, when the driving amplitude of the second sound is sufficiently high, multiple harmonics of second sound waves are generated over a wide range of frequencies due to nonlinearity. At sufficiently high frequencies the nonlinear transfer of the wave energy to sequentially higher wave numbers is terminated by the viscous damping of the waves.