The first consideration should be to remove the cause of the shade! e.g.
If the shade is caused by trees, then thin them out.
Alternatively pollard them or better still remove* the trees if this is practical.
Grass / Turf will rarely do well under trees, it will always be competing for water and nutrients, plus drips from the tree canopy can damage the lawn.
* If removal is an option ensure that there is not a TPO (Tree preservation order) on the tree/s.
If this not an option, then the type of grass will have to be considered.
Most grass / turf require four to six hours of full sun each day, to allow photosynthesis to occur.
If starting from new or renovating choose a shade-tolerant seed mix.
These are often a mix of;
Creeping fescue to bind the lawn together.
Meadow grass for hard wearing and drought tolerance.
Chewing fescue will tolerates close mowing.
Bent grass will give the lawn a tighter appearance and provides good winter greenness.
Always remember even shade tolerant grasses need some light to survive.
If renovating, over-seed the lawn in early September Week 35>.
If you find that the soil is compacted, or of low fertility and heavily infested with moss it may be better to replace it.
Altenatively, you could consider doing away with the lawn altogether, and replacing it with materials such as gravel, timber or concrete decking.
Mow less often, keep the mower blades relatively high, say 50-75mm (2”-3”) high and always lift the clippings.
When watering remember you will be watering the trees at the same time so it is best to water heavily at infrequent intervals, to encourage tree roots to go deeper for their moisture needs.
Reduce the amount of traffic on shaded areas.
You may find that moss will tend to out-compete the grass, so give the lawn an application of Autumn Feed and Weed with added moss killer around Week 35-40, and a Spring Feed and Weed around Week 15-20 the following year.
Prior to applying the spring feed scarify the lawn with a lawn rake or mechanical scarifier to remove any thatch.
Shade cast by buildings, has been left down to the individual to resolve, perhaps he/she could consider alternative ground cover.