Water vapour continuum absorption is an important contributor to the Earth's radiative cooling and energy balance. Here, we describe the development and status of the MT_CKD (MlawerTobinCloughKneizysDavies) water vapour continuum absorption model. The perspective adopted in developing the MT_CKD model has been to constrain the model so that it is consistent with quality analyses of spectral atmospheric and laboratory measurements of the foreign and self continuum. For field measurements, only cases for which the characterization of the atmospheric state has been highly scrutinized have been used. Continuum coefficients in spectral regions that have not been subject to compelling analyses are determined by a mathematical formulation of the spectral shape associated with each water vapour monomer line. This formulation, which is based on continuum values in spectral regions in which the coefficients are well constrained by measurements, is applied consistently to all water vapour monomer lines from the microwave to the visible. The results are summed-up (separately for the foreign and self) to obtain continuum coefficients from 0 to 20 000 cm(-1). For each water vapour line, the MT_CKD line shape formulation consists of two components: exponentially decaying far wings of the line plus a contribution from a water vapour molecule undergoing a weak interaction with a second molecule. In the MT_CKD model, the first component is the primary agent for the continuum between water vapour bands, while the second component is responsible for the majority of the continuum within water vapour bands. The MT_CKD model should be regarded as a semi-empirical model with strong constraints provided by the known physics. Keeping the MT_CKD continuum consistent with current observational studies necessitates periodic updates to the water vapour continuum coefficients. In addition to providing details on the MT_CKD line shape formulation, we describe the most recent update to the model, MT_CKD_2.5, which is based on an analysis of satellite- and ground-based observations from 2385 to 2600 cm(-1) (approx. 4 µm).