In this talk I will give some examples of recent research in our group on the modelling and simulation of patients with respiratory diseases, focussing in particular on chronic obstructive pulmonary disorder (COPD) and acute respiratory distress syndrome (ARDS). I will show how mathematical and computational techniques can be used to advance research into the development of new treatment strategies for these diseases, by uncovering mechanistic links between disease pathophysiology, therapeutic interventions, and patient outcomes.