The observational study of transiting exoplanet atmospheres began in the early 2000s and is flourishing today. I will review the current sample of known exoplanets and the techniques used to characterise their atmospheres. Until now, much of the atmospheric characterisation work has focused on the extreme population of hot Jupiters and I will describe a few notable results. Due to the more challenging nature of the observations, progress has been relatively limited for planets Neptune-sized-and-smaller, but this is set to change as we enter the JWST era. I will preview some of the advances that we are anticipating for the characterisation of rocky planets and sub-Neptunes using this revolutionary facility over the coming years. Those unable to attend the colloquium in person are invited to participate online through Zoom (Meeting ID: 942 0262 2849, passcode 792771) using the link: https://zoom.us/j/94202622849?pwd=dGlPQXBiUytzY1M2UE5oUDRhbzNOZz09