Cancer, a complex and multifaceted disease, is characterized by distinct biological capabilities known as the Hallmarks of Cancer. These hallmarks include sustaining proliferative signalling, evading growth suppressors, resisting cell death, enabling replicative immortality, inducing angiogenesis, activating invasion and metastasis, and more recently, avoiding immune destruction, deregulating cellular energetics, and promoting genome instability.

Similarly, Mathematical Oncology has, since its inception 50 years ago, focused on studying these hallmarks through various mathematical approaches. These include differential equations, agent-based models, and stochastic processes. These efforts aim to quantify and predict the dynamics of tumor growth, metastasis, treatment, and therapeutic responses, thereby providing a deeper understanding of Cancer Biology and informing treatment strategies.

In this talk, we will explore major developments in the mathematical study of the Hallmarks of Cancer, tracing their evolution from initial conceptualization to the current understanding of Cancer Biology and their contributions to Medicine. During this process, we will employ various mathematical approaches and demonstrate the utility of these models. By integrating biological data, these models help elucidate the dynamic interactions between cancer cells and their microenvironment. Specific emphasis will be placed on predicting tumor growth, invasion, metastasis, treatment, and the emergence of resistance.

We will conclude with the most recent developments in the field and discuss a number of open questions that remain open for the future of Mathematical Oncology. These will include potential new directions for research, emerging techniques in modeling, the use of Artificial Intelligence and Virtual Reality techniques, and the ongoing challenges in translating mathematical insights into clinical practice and ultimately contributing to improved patient outcomes and Personalised Medicine.