I will present new results on sharp decay estimates and asymptotics for small solutions to 1D Klein-Gordon equations with variable coefficient quadratic and cubic nonlinearities. The need for a precise understanding of the asymptotic behavior of such solutions for instance arises in the asymptotic stability analysis of ”kink” solitons in one-dimensional field theories. In this talk I will first describe the difficulties caused by variable coefficients to analyze the long-range nature of quadratic and cubic nonlinearities in 1D. Then I will explain a new approach based on local decay estimates to overcome these issues in the variable coefficient cubic case. Subsequently, I will also discuss the variable coefficient quadratic case, which exhibits a striking resonant interaction between the spatial oscillations of the variable coefficient and the temporal oscillations of the solutions. This is joint work with Hans Lindblad and Avy Soffer.