Delivery of drug formulations through the subcutaneous route is a widely used modality for the treatment of several diseases, such as diabetes and auto-immune conditions. Subcutaneous injections are typically used to inject low-viscosity drugs in small doses. However, for new biologics, there is a need to deliver drugs of higher viscosity in large volumes. The response of subcutaneous tissue to such high-volume doses and higher viscosity injections is not well understood. Animal models have several drawbacks such as relevance to humans, lack of predictive power beyond the immediate population studied, cost, and ethical considerations. Therefore, a computational framework that can predict the tissue response to subcutaneous injections would be a valuable tool in the design and development of new devices. To model subcutaneous drug delivery accurately, one needs to consider: (a) the deformation and damage mechanics of skin layers due to needle penetration and (b) the coupled fluid flow and deformation of the hypodermis tissue due to drug delivery. The deformation of the skin is described by the anisotropic, hyper-elastic, and viscoelastic constitutive laws. The damage mechanics is modeled by using appropriate damage criteria and damage evolution laws in the modeling framework. The deformation of the subcutaneous space due to fluid flow is described by the poro-hyperelastic theory. The objective of this review is to provide a comprehensive overview of the methodologies used to model each of the above-mentioned aspects of subcutaneous drug delivery. We also present an overview of the experimental techniques used to obtain various model parameters.