We consider the Westervelt equation in an unbounded domain and propose nonlinear absorbing boundary conditions for its efficient and robust numerical simulations. We use the theory of pseudo- and para-differential operators as well as asymptotic expansions to derive local in space and time absorbing boundary conditions of low to high orders in a consistent way. We show that the pseudo- and para-differential theories lead to essentially the same absorbing boundary conditions in terms of computational efficiency and numerical accuracy, whereas the asymptotic expansions result in exactly the same boundary conditions as the ones obtained with the para-differential approach. Moreover, we demonstrate that the use of pseudo- and para-differential operators leads to the same boundary conditions if the nonlinear function to be linearized vanishes at zero. The numerical studies demonstrate both the efficiency and effectiveness of the developed boundary conditions for different regimes of wave propagation in a wide range of excitation frequencies and angles of incidence.