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Prune mature bushes in winter, cutting out all three year old wood (and horizontal and downwards growth) as close to the base of the plant as possible. It's quite easy to see the difference between first, second and third year wood. First-year wood is smooth, and warm brown in colour, second-year wood is rougher and grey, and third-year wood is rough and darkest of all. Once stems are two years old or more, they are not as productive as younger wood.
With extremely vigorous bushes, you can also prune out up to a third of second-year shoots as well, but these are best retained in less vigorous bushes. The aim of pruning is to have upright, healthy stems, preferably growing from ground level, being replaced on a gentle, annual cycle of pruning.
A tip for blackcurrants is that third-year wood can be cut out when harvesting the ripe berries too. This makes harvesting easier, and leaves you with less pruning work in winter to worry about. You'll then only need to tidy up the remaining growth following the instructions above.
RJB