Live in London, overall safe.
Live in a 'safe' part of London, St. John's Wood, but have still have experienced an attempted mugging, not far from my home (not violent and not successful).
Being shot is not something I foresee, but then again I don't use the busses.
Someone brought up a valid point, there are social housing estates in even the best areas, Mayfair, Chelsea, St. John's Wood, Kensington, etc. So you're probably as much at risk in the 'better areas' because the criminals know who is worth targeting. I could be wrong though, you do hear of more severe crime in Hackney, Haringey, Islington, Clapham, Brixton, Walthamstow, Shoreditch, Stockwell, or Dalston than Chiswick, Hampstead, Highgate, Belgravia, Knightsbridge, Maida Vale, or Richmond.
Yet, to say that crime is high in London is absurd, its one of the world's largest cities and there will be incidents.