Aims. The Atacama Large Millimeter Array (ALMA) is an international astronomy facility to be used for detecting and imaging all types of astronomical sources at millimeter and submillimeter wavelengths at a 5000-m elevation site in the Atacama Desert of Chile. Our main aims are: describe the correlator sub-system which is that part of the ALMA system that combines the signal from up to 64 remote individual radio antennas and forms them into a single instrument; emphasize the high spectral resolution and the configuration flexibility available with the ALMA correlator. Methods. The main digital signal processing features and a block diagram of the correlator being constructed for the ALMA radio astronomy observatory are presented. Tables of observing modes and spectral resolutions offered by the correlator system are given together with some examples of multi-resolution spectral modes. Results.The correlator is delivered by quadrants and the first quadrant is being tested while most of the other printed circuit cards required by the system have been produced. In its final version the ALMA correlator will process the outputs of up to 64 antennas using an instantaneous bandwidth of 8 GHz in each of two polarizations per antenna. In the frequency division mode, unrivalled spectral flexibility together with very high resolution (3.8 kHz) and up to 8192 spectral points are achieved. In the time division mode high time resolution is available with minimum data dump rates of 16 ms for all cross-products.Read less <