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Step-free access to tube stations is something that TfL and Sadiq Khan take very seriously. There are 78 tube stations on the network with step-free access; as well as 60 London Overground stations; and 9 TfL Rail stations also with step-free access. The aim is to make another eight tube stations step-free by spring 2022.
The DLR is completely step-free and the Elizabeth line will be, when it's completed.
However, there are plenty of stations that don't quite qualify as step-free, but are still usable for those who can't handle stairs, be that for disability or health reasons, travelling with buggies or luggage, or just a prejudice against steps. The precise details of which stations you can avoid stairs at are far too complex to fit on the normal tube map, which is why the Avoiding Stairs Tube Guide exists.
The map uses a plethora of multicoloured rings to explain just how step-free each station is. It details which stations are only step-free in one direction (surprisingly quite a few), which ones have level access, where you can change between lines without using stairs, and even mentions which stations are accessible using only a small number of steps.
The guide doesn't just exist online either. You can pick a physical copy up from one of TfL's Visitor Centres dotted around London. In the meantime, take a look at the guide online [pdf].