What's the most likely cause of damp in my ground floor flat walls?

Failed damp proof course, the course is being bridged, or (most likely) no dpc at all. Damp proof courses are often absent in 19th century buildings but when they were heated with coal fires tended to be far better ventilated than today so it wasnt so much of a problem. Make sure nothing is left up against the wall on the outside that could prevent things drying out and make sure air bricks are clear if you don't want the floor boards to be affected.

The solution is usually an injected dpc (electric ones don't work in my opinion) or insertion of physical one which is much mor difficut to do, but you should refer the problem to the landlord as it is likely you don't own the walls. Check your lease, though, the cost for any work may be passed back to the tenants. If it was a conversion find out if there was any damp proofing done, but my experience is that any dp guarantees are worth less than the paper they are written on. There are tell-tale holes about 10mm in daimeter drilled all along the wall just above ground level.

I agree with the previous answers that it is most likely a damp course failure and will probably need to be expertly repaired. However I successfully cured an outside loo from this problem with a product called Aquaseal Waterproof Tanking for here: http://www.thf-tradesupplies.co.uk/aquaseal-waterp... It might not be suitable for using throughout the entire flat, but its an idea.

It's a brick house

Your damp coursing has failed. If it is a rental then contact the landlord to fix it (they are legally bound to do so, but you may put yourself at risk of a retaliatory eviction depending on how big a piece of scum the landlord is), if it is your own then contact your insurer, it should be covered under building's insurance.

Well the moisture from the soil is coming through the brick. There is no water proofing in this old house. You would have to trench around the oitside and add a water proof layer. In north America they use cement foundations treated with tar-like products and insulated with styrofoam products. Big job. Can you live with just a trench and redirect roof water away from the foundation?