Numerous areas of mathematical oncology are explored including the phases of avascular tumour growth, vascular tumours, tumour growth In Vitro, drug infusion, heterogeneous growth of solid tumours and prostate cancer. Three phases of avascular tumour growth are considered and the laws of diffusion are applied to model their growth. For the first phase integration is used to find the time it takes for a proliferating tumour to reach the second phase of growth, where a quiescent core forms. For modelling the second phase of growth boundary conditions are applied to obtain the oxygen concentration as a function of the tumours radius. Boundary conditions for vascular tumours are explored where the tumour forms a connection with the blood or lymph vessels. Studying tumour growth In Vitro is important as it allows mathematical analysis of scientific experiments carried out in test tubes. A simple model for Continuous drug infusion is studied and then a more complex and practical model of periodic infusion is looked at. A competition model for tumours which have two competing phenotypes of cancer cells is analysed. For this heterogeneous growth it is valuable to look at the stability of eigenvalues by applying the trace-determinant method on a 3-by-3 matrix in order to determine which cancer cell phenotype will dominate. Finally, a model for prostate tumour growth and androgen independent relapse is explored. Bifurcation diagrams allow us to see how changing certain parameters effects the success of prostate cancer treatments such as Androgen Deprivation Therapy(ADT).