In this paper, the 𝐺′/𝐺 expansion method is applied to the (2+1)-dimensional Asymmetric-Nizhnik-Novikov-Veselov equation (ANNV). The motivation is creating new families of solitary waves. The system of equations has been combined in one partial differential equation (PDE) and the traveling wave variable has been applied to transform the resultant equation into an ordinary differential equation (ODE). The homogenous balance condition has been applied to determine the truncation variable of the 𝐺′/𝐺 expansion. Four cases are created according to the appropriate choice of the arbitrary constants relations. For each case, some new solitary wave solutions including solitons and kinks represented by trigonometric, hyperbolic, logarithmic, polynomial, and combinations of these functions.