This thesis illustrates the applications of diffusion equations in the biological and medical sciences. The modelling of the avascular stage of tumour growth is discussed here. Three models are considered to study the evolution of the tumour in its early life's cycle. The first model is predicted the maximum pre-angiogenic tumour size. The second model is estimated the upper bound of the radius of tumour before it enters the second stage, vascular stage. The third is a more realistic model, and it is used to study the evolution of the tumour with time. The strengths and the weaknesses of each model are discussed in this thesis.