The A3217 runs from London Victoria Station into Chelsea.
The road starts in the one-way-system mess by Victoria station - the exact spot depends on which map you're looking at with Lower Grosvenor Place either being part of the A3217 or A302. Either way, the road leaves the one-way system at traffic lights on the A302 Grosvenor Place, on the edge of Buckingham Palace Gardens, and heads southwestwards.
It crosses Eaton Square, where it meets the A3213 in the middle, to run into Sloane Square, which is a roundabout, where it crosses the A3216. This marks the start of Chelsea's King's Road, which the A3217 follows for the rest of its route - which is about twice as long as what has gone before.
The road crosses Edith Grove, the southbound A3220, at traffic lights, before ending at the next set of lights on the northbound A3220. The road ahead is the A308 and the number changes for no apparent reason.
Originally the route of the A308 was not as bizarre as it is now and so the A3217 continued west along the whole length of the King's Road, to end on the A219 on the north side of Putney Bridge. However, the original eastern end of the A3217 was at Sloane Square. The road east from here to Victoria (which did include Lower Grosvenor Place) was the B311. Tracing the history of roads in London is always difficult but it seems likely that the A3217 was extended to the east before being truncated in the west.