NB, please do not rely solely on this information for your travel planning.
THE UK: BY AIR
Within Europe: carriers include British Airways, easyJet, Ryanair, Air Berlin, Vueling, Air France, Norwegian, Transavia, Air Italy, Wizzair, Flybe, TUI, TUI Airways, BMI and Thomas Cook.
From the USA: try British Airways, WOW air, Delta, American Airlines, Virgin Atlantic and United Airlines.
FROM THE AIRPORT
The Heathrow Express train whisks you into London Paddington in about 15 minutes but it's quite expensive; a much cheaper and slightly slower alternative is the Heathrow Connect (25 minutes into Paddington). There are underground (Tube) links into London too, as well as coach services (mostly National Express) to cities across the country. The Gatwick Express train shuttles straight into London Victoria in about 30 minutes; there are stopping trains too (slower and cheaper), which also serve the south coast (Brighton etc) and south west (Reading etc). The Stansted Express train runs into London Liverpool Street in 45-60 minutes, stopping at various stations en route; and there are coaches via northeast London to Victoria, as well as trains direct to East Anglia and the Midlands. Luton has fewer links.
If you want to hire a car, see our car rental recommendations. If driving into central London, you will need to pay the daily 'congestion charge' (if hiring a car, ask about this in advance).
British Airways flights connect the major cities (London, Manchester, Leeds, Newcastle, Glasgow, Edinburgh etc), while budget carriers Ryanair and Easyjet are also useful if travelling from the southeast (Stansted/Luton) to Scotland (Prestwick, Edinburgh) or northeast England (Newcastle).
For cheapest fares, try and book your ticket as far in advance as possible, using thetrainline.com or nationalrail.co.uk. Finding the cheapest fare can be astonishingly complex - it may even be 2 singles. Some routes, especially down the west of England (e.g. Bristol to Leeds / Newcastle), are run by Virgin Trains. If your train arrives over an hour late, you are normally entitled to compensation, but you will have to fill in a form with details and tickets, and give a UK address for the refund.
GETTING AROUND LONDON
By Tube (Underground)
Extensive and easy to use, and popular routes now run at night. Get an Oyster card at any tube station - journeys cost almost half the normal cash fare. See the Oyster website. You can also use Contactless (check with your bank) instead of an Oyster.
By day, hail a black cab - the distinctive black car which seats up to 5, and where journeys are metered.
There are still some red double decker buses - sitting on the top deck is a great way of seeing the city. There are also some night buses on popular routes (preceded by 'N') if you need to get home cheaply after about 11.30.