There are, perhaps suprisingly, a good number of perennials that grow well in shady conditions.

For a dry, shady spot such as near a hedge or under the canopy of a large, evergreen tree, Alchemilla mollis with delicate sprays of lime-green flowers in summer, Iris foetidissima - evergreen, sword-shaped leaves and showy bright red seeds in pods throughout the winter, and Symphytum (Comfrey) will all thrive.

With more moisture the range increases, here is a small selection:
Bergenia with large evergreen leaves and spikes of flowers - either pink or white - in spring and then often repeated later in the year. The variety 'Silberlicht' is particularly good, the white flowers shine in a dark corner.
Epimediums make excellent ground cover, the heart-shaped leaves, tinted bronze when young, suppress weeds well.
Geranium maccrorhizum with hairy aromatic leaves and, depending on variety, flower colour white to magenta, spreads neatly and is ideal for a cottage garden style planting.
Hostas, with their attractive foliage, are best suited to pot culture, away from slugs and snails, They look magnificent in a shady spot on the patio.
Primulas are a large family and there are many that grow well in some shade; two examples - Primula florindae, the Giant Cowslip, with large yellow bell-shaped flowers thrives in full shade, Primula 'Garryade Guinivere' with maroon coloured leaves and complimentary lilac flowers likes partial shade.
Pulmonaria give welcome colour in the early spring and their silvery, spotty leaves continue to give interest. If the leaves begin to look a bit tired they may be cut back to give a second fresh flush in summer.

And the ferns all like shade too...