The grinding and polishing machine has a tool-carrying spindle (1) driven with a combined rotating and rocking action. It has a driving shaft (4) which is reciprocated axially and passes through a disc (8) which slopes at an acute angle. Above this disc is a second disc attached to the shaft, with a roller (5) attached to its periphery which rotates against the sloping disc. The tool spindle also passes through another sloping disc and through a plate (12) with a peripheral roller (16) which travels against this second sloping disc. The tool spindle carries an arm (13) with a dog projecting from it to fit into a hole in the plate to rotate the same. This plate also has a dog (14) which fits into a hole (15) in the disc attached to the driving shaft so that it transmits rotary movement for driving the tool spindle.
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