The glial cell is the most numerous cell type in the central nervous system and is believed to play an important role in guiding brain development and in supporting adult brain function. One type of glial cell, the astrocyte also may be an integral computational element in the brain since it undergoes neurotransmitter-triggered signalling. Here we review the role of the astrocyte in the central nervous system, emphasizing receptor-mediated Ca2+ physiology. One focus is the recent discovery that the neurotransmitter glutamate induces a variety of intracellular Ca2+ changes in astrocytes. Simple Ca2+ spikes or intracellular Ca2+ oscillations often appear spatially uniform. However, in many instances, the Ca2+ rise has a significant spatial dimension, beginning in one part of the cell it spreads through the rest of the cell in the form of a wave. With high enough agonist concentration an astrocyte syncitium supports intercellular waves which propagate from cell to cell over relatively long distances. We present results of experiments using more specific pharmacological glutamate receptor agonists. In addition to describing the intercellular Ca2+ wave we present evidence for another form of intercellular signalling. Some possible functions of a long-range glial signalling system are also discussed.