A glacier is treated as an essentially one-dimensional flow system, which is, continuously along its length, either gaining new material by snowfall or losing it by melting and evaporation. The flow at each point, i.e. the volume passing a given point per unit time, is assumed to be a function of the thickness and surface slope. It is then shown that a region of uniform longitudinal compressive strain-rate is temporarily unstable. Lighthill & Whitham's theory of kinematic waves is applied to study the response of a simple model to a sudden chango in rate of accumulation (snowfall). All parts of the glacier thicken, but the lower parts thicken unstably until a kinematic wave arrives to restore stability. The thickening of the lower parts, and the advance of the glacier, can be very great for only a small change in accumulation; the analysis thus explains why glaciers are such sensitive indicators of climate. The velocity and diffusion of kinematic waves is discussed; they travel at 2 to 5 times the surface speed of the ice. Any part of a glacier or ice-sheet has a characteristic response time to changes in accumulation-about 5000 yr for the Antarctic ice-sheet as a whole, and 3 to 30 yr for a typical glacier. The amplitude and phase of the response of a simple glacier to high frequency (seasonal) and low-frequency (climatic) periodic changes in accumulation is calculated. (The periodic changes in accumulation are the Fourier components of the complicated variation that actually occurs.) The upper part behaves in a simple way; the lower part, on the other hand, shows not only a direct response, but also a delayed response due to the arrival of kinematic waves from the upper part. The two components interfere and give wide scope for variation in behaviour from one glacier to another. A further instability, discussed by Bodvarsson, which arises from an increase of accumulation with altitude, is included in the theory. It is also shown how the theory is to be applied in a valley of changing width to calculate the development of any disturbance produced by climatic change. Thus the theory enables one to calculate, in principle, the response of any glacier to any climatic change.