Weakly nonlinear propagation of diffracted sound beams in nonuniform bubbly liquids is theoretically examined. The spatial distribution of the number density of the bubbles, initially in a quiescent state, is assumed to be a slowly varying function of the spatial coordinates; the amplitude of variation is assumed to be small compared to the mean number density. Two types of nonlinear wave equations for progressive quasi-plane beams in weakly nonuniform bubbly liquids are then systematically derived via the method of multiple scales. The diffraction effect is incorporated by adding a relation that scales the circular sound source diameter to the wavelength into the original set of scaling relations composed of nondimensional physical parameters. A set of basic equations for bubbly liquids is composed of the averaged equations of mass and momentum in a two-fluid model, the Keller equation for bubble wall, the equation of state for gas and liquid, the mass conservation equation inside the bubble, and the balance equation of normal stresses across the gas-liquid interface. As a result, two types of evolution equations, a nonlinear Schrödinger (NLS) equation including dissipation, diffraction, and nonuniform effects for high-frequency short-wavelength case, and a Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation including dispersion and nonuniform effects for low-frequency long-wavelength case, are derived from the basic set. Finally, numerical and analytical solutions of NLS and KZK equations toward some applications are presented.