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Footings subjected to dynamic loads are commonly designed under the simplifying assumption of linear or equivalent-linear viscoelastic soil behaviour. Even though this approach is simple to implement and, in some cases, could take advantage of available closed-form solutions, the outcomes remain a gross approximation - especially for high excitation amplitudes such as those induced by strong earthquakes. Although considerable research has been conducted for the case of large footing vibrations, where uplift, slippage or other types of foundation failure may occur, there remains a research gap for small to medium strain amplitudes (levels of strain between $10^{-6}$ and $10^{-3}$), for which the soil behaviour is also non-linear, but not close to failure. To address this problem, a numerical methodology is developed herein, for the analysis and design of shallow footings, while taking into consideration shear modulus degradation and hysteretic damping increase effects for the foundation subsoil. Non-linearities such as hysteresis and rate dependence are the main characteristics of cyclic soil behaviour. These features prove the frequency domain analyses inadequate, and more computationally demanding time domain analyses, such as those conducted in this study, are more appropriate. The analysis methodology is based on the implementation of two numerical models, namely the Ramberg-Osgood (RO) model and the Modified Hyperbolic model (MH), as user-defined models into the explicit finite-difference code FLAC. Focus is then given on a rigid strip surface foundation subjected to harmonic oscillations, and results are presented in terms of the variation of the dynamic impedance with the dimensionless frequency of excitation. In addition, results are provided for the corresponding three-dimensional problem of a square footing resting on the surface of a soil layer over rock. Linear elastic static and elastodynamic available solutions are first revisited and comparisons are drawn against published results following a throughout parametric investigation. Vertical, horizontal and rocking oscillations are considered, with the excitation imposed in the form of an applied harmonic displacement or a rotation atop a rigid surface strip footing. Non-linear soil behaviour is then studied, and different excitation amplitudes are considered to explore the effects of soil non-linearity in terms of governing parameters such as dimensionless depth to rock, soil plasticity index, material damping, and Poisson’s ratio. Strain rate effects are also examined. The analysis is then extended to three dimensions and comparisons are made between 2-dimensional and 3-dimensional response in all excitation modes. Results are presented in the form of dimensionless equations, graphs and charts which are suitable for use in geotechnical engineering practice. A detailed discussion on the use of Rayleigh damping in problems of this type is provided. Finally, a case study involving field tests in a test site in California is modelled and the predictions are compared to the experimentally obtained data.