Modelling the response of Himalayan glaciers to rapid climate change is an important problem. The poorly understood effects of debris cover and the variable response of the glaciers have made it difficult to understand their dynamics. We propose a simple model for debris-covered glaciers and validate it against data from Dokriani Glacier, India. Numerical investigations of the model show that the response of debris-covered glaciers to a warming climate has two timescales. There is a period when the glacier loses ice by thinning but the front is almost stationary and it develops a long, slow-flowing tongue. This stationary period, which can be >100 years for glaciers with a large extent of debris cover, is negligible for bare glaciers. The quasi-stagnant tongue does not develop in response to cooling. An analysis of remote-sensing data in the light of these results indicates that the variable response of the glaciers in the Himalaya is consistent with a climate that is warming on average, but has considerable spatial variability in the warming rates. We estimate the average warming rate to be about the same as the global average.