Richards' equation, the governing equation for single phase unsaturated flow in soil, is highly nonlinear and, consequently, is difficult to solve either analytically or numerically. The nonlinearity enters through the hydraulic functions, viz., the soil moisture characteristic curve and the unsaturated hydraulic conductivity. We present an exact quasi-analytical solution for the one-dimensional Richards' equation subject to an arbitrary time-dependent ponding depth. The solution depends on solving a first-order ordinary differential quation. For certain special cases, such as when the surface ponding depth is constant, fully analytical results can be derived. The theory is used in a number of applications. For example, an existing analytical approximation for cumulative infiltration subject to a ponded, time-dependent surface head is checked. As well, a drainage formula is derived. New results for solute transport coupled with a nonlinear adsorption isotherm are derived by making use of an exact mapping between Richards' equation and the governing solute transport equation.