If you hope to defend yourself against violent attacks, studying martial arts is a great way to improve your skills and build muscle memory. However, there are a few things you'll find in martial arts training sessions or matches that you won't find in real-life self-defense situations.
Safety equipment is an obvious omission — the guy who's trying to steal your wallet or assault you in a drunken rage won't be wearing padded gloves, and if you get knocked down, it won't be on a cushy mat. The other factor in real street fights is the total absence of rules. There's no referee and no holds barred. Groin strikes, eye gouging, throat punches, and even weapons can easily come into play.
So, while martial arts training can — and arguably should — be used to improve your self-defense skills, it must also be translated into real-world context. For example, many schools of martial arts such as Brazilian jiu-jitsu teach grappling and ground fighting. That's fine for regulated matches, but in a real-life street fight, it can go wrong quickly. One head slam against concrete, and it's lights out.
For more tips on real-world ground fighting, check out Michael's article, Ground Fighting Realities: How Not to Get Stomped Into a Stain.