Contactless Payment on London Transport

Contactless:

Since 2014 it has been possible to use a contactless method of payment wherever you can use an oyster card, which is the entire tube network and much of the London sections of the national rail system along with the bus, Tube, tram, DLR and Emirates Air cable car. This means you can use your normal debit or credit card for payment and still benefit from the daily caps that oyster cards carry and even get a 7 day (Monday to Sunday) cap that is not available when using an Oyster card. Please note that this is not the same time period as a 7 day Travelcard which runs between any 7 days and not tied to Monday to Sunday.
Contactless means any payment method using NFC (near field communication) and includes smart phone payment systems. If you have a contactless payment card issued in the UK you will be fine using it for payment (if it supports NFC, look for the symbol). Overseas visitors may still be able to use their cards.
Currently all American Express contactless cards can be used and nearly all MasterCard and Maestro contactless payment cards issued outside the UK are accepted. Some Visa and V PAY contactless payment cards from countries issued outside the UK are not accepted.Also watch for overseas transaction charges which may apply for travel made with a card issued outside the UK.
If you use a pre-paid card you will also need to check if this will be accepted on London transport.
Main Benefit:
The main benefit for contactless payment is there is no need for loading a card with credit beforehand and Monday to Sunday capping. However the point to note here is that any caps are linked to a single payment method, so you cant use more than one card for example and have all payments count towards capping. Always use the same method for payment for each journey to have those payments grouped together for capping purposes.

Receipts and Tracking:

If you sign up for a TFL account you also benefit from having details of fares charged and payments for all your journeys and up to 12 months of detailed journey history, useful for demonstrating business travel costs for example.

Downside:

One big drawback of the system is that it does not allow for use by more than one passenger at a time. For example you cant pay for a group of friends and benefit from price capping. Also discounts can not be added to contactless payment cards which Oyster cards can do. Additionally you cannot load other forms of tickets such as Travelcards onto a Contactless card as you can with an Oyster card.

Very Important (Card Clash):

Be careful of what has been termed card clash. It is very important that when you touch in and touch out (buses and trams only require touch in) that you use the SAME card. Using different cards for touch in and touch out will result in your being charged the maximum amount for that journey TWICE. This can happen if you have two cards in close proximity to the reader when touching in or out.

Re-cap:
Oyster and Contactless are different methods of payment. Read full details of Oyster payment here.
Oyster fares and Contactless fares are identical EXCEPT for:
Oyster cards accept travel discounts, Contactless cards do not.
Contactless cards have a Monday to Sunday capping discount not available with an Oyster card. Hopper fares:
Make a journey using any form of contactless such as debit and credit card or Oyster on a bus or tram, and if you make another journey on a  bus or tram within one hour the second journey is free. Journey time is calculated from the touch-in times of payment. Make sure you touch-in for the second journey. The second free ride is automatically applied and nothing is else is required.
Exceptions:
Using the Tube, DLR, London Overground, TfL Rail, Emirates Air Line, River Bus or National Rail services between the first and second bus or tram journey means you must pay in full for the second journey.
If your Oyster card has a negative pay as you go balance after the first journey you wont get the free ride. However if you clear the negative balance within one hour of touching in, you can still make the second journey for free
You cannot use an Oyster card to change from a tram to a bus at Wimbledon station. For these journeys, an automatic refund for the second fare will be paid to your Oyster card.

2018 Price comparison
Tube, London Overground and DLR: