When we construct mathematical models to represent a given real-world system, there is always a degree of uncertainty with regards to the model specification - whether with respect to the choice of parameters or to the choice of formulation of model functions. This can become a real problem in some cases, where choosing two different functions with close shapes in a model can result in substantially different model predictions. This phenomenon is known as structural sensitivity, and is a significant obstacle to improving the predictive power of models - particularly in fields where it is not possible to derive the functions suitable for representing system processes from theory or physical laws, such as the biological sciences. In this talk, I shall revisit the notion of structural sensitivity and propose a general approach to reveal structural sensitivity which is a far more powerful technique than the conventional approach consisting of fixing a particular functional form and varying its parameters. I will demonstrate that conventional methods based on variation of parameters alone will often miss structural sensitivity. I shall discuss the consequences that structural sensitivity and the resulting model uncertainty may have for the modelling of biological systems. In particular, it will be shown the concept of a 'concrete' bifurcation structure may no longer be relevant in the case of structural sensitivity, thus we can only describe bifurcations of completely deterministic systems with a certain probability. Finally, I will show that structural sensitivity can be a possible explanation of the observed irregularity of oscillations of population densities in nature. At the end, we will discuss the current challenges related to structural sensitivity in models and data.