Physics-Informed Neural Networks (PINNs) have emerged as a promising meshless framework for solving partial differential equations, embedding physical laws directly into the training process of neural networks. Their potential applicability to mathematical biology is particularly appealing, as many biological phenomena are governed by complex, nonlinear PDEs defined on irregular domains where classical discretisation methods can be costly or cumbersome. This thesis investigates the capabilities and limitations of PINNs when applied to biological equations, with a focus on systems that exhibit spatial pattern formation. Following a thorough theoretical treatment of the PINN framework, including loss formulation, network architecture, training dynamics, and known failure modes, two case studies of increasing complexity are examined.