I presume you want year round privacy so evergreen plants are the answer.

The obvious one is Leylandii, but they can be a bit of a thug so maybe choose Laurel or Grisellinia. Eleagnus, Eucalyptus (another thug), Garrya, Thuja, Arbutus and pittosporum tenuifolium would all work

A photo of the area would be useful - if the garden is small, you don't want something that will shade the whole garden or bush out too far. Thuja 'Emerald Green' gets 12-14 feet with a spread of 3-4 feet - it's not as fast growing as Thuja Green Giant, but the trouble with fast growing plants is, they don't stop at a reasonable height - it gets 50 feet and you'll need to keep it cut every year, so just as much of an unwelcome nuisance as the accursed Leylandii cypress.

The above and Photinia. Eucalyptus are instant trees - some grow from seed to 7ft in 2 years. You can get the same speed with Rowan wips but they are not evergreen.

You want trees, but your neighbours at the bottom of your garden may not want your trees, with the shade they give them, the leaves they will have to clear up in the autumn and the seeds that will start to grow in their garden. Also the roots will be under their garden possibly creating further problems. It's a particular 'achilles heel' to me as I don't know why people want to plant trees right next to a boundary, wall or fence, in urban gardens. If you live in an area with no neighbours around you then that is fine.It is plain selfish in the middle of a housing estate . A 6ft fence or shrubs will give all the privacy that you need.

That's true. We lived some years ago next door to a very tall leylandii hedge which took all the goodness and most of the light form the border. A fence with a trellis on top with some climbers should work just as well and be more neighbourly and quicker. There are often questions on here from people who have next door's trees leaning over their gardens, with shade from the branches and dry ground from the roots.

Plant fast growing evergreen trees by all means - just be prepared for the complaints from your neighbours and the effort that you will have to put in to keep them on the right side of the high hedges legislation.
A 6ft fence is more than adequate for privacy but you can add trellis with climbing plants if you really must have something above head height.

Doreen makes a good point but to me privacy is more important than neighbours wish list.

"A 6ft fence is more than adequate for privacy"

Not if your plot is on a slope.

I don,t know how long your garden is but if it possible can you make a nice,patio area at the rear of your house and section it off with fence/trellis so you have your instant privacy,without having to wait for anything to grow.Also ,as it is a new build I take it it is a blank canvas,you might ,at a later date,want to put a shed,lawn or other things down there.What I find is anything large growing certainly cuts down your space, kerryvw.

Good thinking Kerry.