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Can be extended to include Fulham Broadway route.Total about 2 ½ miles. Note:The cemetery now has a cafe and visitor centre. 

Brompton Cemetery is immediately next to the station (turn right). It was opened in 1840 to help relieve London of the problems caused by unhealthily overcrowded churchyards. Created on 39 acres of land, most of which had been fields and market gardens, the original design was on a grand scale, with colonnades and a domed chapel (based on St. Peter’s in Rome, no less). 
Unfortunately all this proved more expensive than expected, and the project had to be pared down.  The construction of the scary catacombs with great iron doors turned out to have been skimped, and they were never a commercial success, being little used. None of this is apparent when you visit this fascinating place with its mature trees and intriguing monuments.  As you can see from the notices at the entrances, it was always the intention that people should be free to stroll about, admire the memorials and perhaps be inspired by the achievements of the many distinguished people who are buried here. Probably because it's now managed by the Royal Parks, it is more attractive than those wretched cemeteries where the dreaded Health and Safety people’s solution to the perceived danger of wobbly monuments is to push them over - resulting in acres of flattened crosses and fallen angels Behind the chapel it is a little wilder, but not depressingly so. Nonetheless, please be careful if you are looking for a particular memorial - you are supposed to stick to the paths.

Many of the most interesting graves are in the central avenue.  Starting from the Old Brompton Road entrance and looking left, you will soon come across a charming sculpture with two small angels, erected to the memory of one Albert Emile Schloss, d.1905. Two trees further on a fine Celtic cross marks the resting place of suffragette Emmeline Pankhurst.  People regularly leave small bunches of flowers here. The tomb of Frederick Leyland, shipowner and patron of the arts is also worth searching out.  Designed by Burne- Jones, its copper is now changed to a glorious green, and the filigree ironwork that encases the casket is in perfect condition. It is on the western side of the central avenue behind a mausoleum, four trees back from a turn-off shortly before you reach the colonnades.

More history and a list of the famous people buried in West Brompton can be found here. (Note that the office is only open by appointment.) To the south west there is a large section for war graves, part of which contains a circle of white stones grouped in rather ghostly fashion round a cross with cannon balls at the base.

Continuing the war theme, an unusual tombstone is that of Sub Lieutenant Reginald Warneford VCRNAS who destroyed a Zeppelin in 1915 - a feat which so impressed the readers of the Daily Express that they contributed to a stone which has a relief of a Zeppelin being bombed. You will find this remarkable monument on the central avenue near the north end of the east colonnade. 
You might want to check that Chelsea F.C. are not playing at home, as West Brompton station is used by some football fans as an alternative to Fulham Broadway. (They take a short-cut through the cemetery.) From the Fulham Broadway link you can discover more things to find in this interesting area.

Photos (click to enlarge)
West Brompton chapel
East colonnade looking north/entrance to catacombs
Notice on main entrance to cemetery
Schloss memorial/Emmeline Pankhurst cross
Leyland memorial/Warneford memorial
War memorial

This is just one walk from the many to be found at London Tube Rambles. There are architectural gems, beautiful country views, historic places and quirky buildings to be found, in the area covered by the Greater London Underground stations. Usually the discoveries are within a mile of the Tube - often only five minutes walk away. If you reached this as an individual page via a search engine, you might like to click on the link above and see the other destinations explored. You'll be amazed at what's out there!