Well, here's the way I've been taught.
We do weapons practice in aikido, and the techniques are often very similar to recieving an empty hands attack - almost identical, really. The emphasis and focus changes slightly, as their destructive potential is increased and their appendages no longer terminate at the ends of their fingers (or toes, I guess, if they have a knife sticking out of their shoe). And of course you want to try to control them so that you are knicked and cut as little as possible. But we're encouraged to treat uke like they have a weapon durring empty hands practice, and to move like we're empty handed when we practice with weapons. This will help to teach you respect for any given person's potential power and deadliness. Rember, you're not fighting and defeating the weapon, you're fighting a person.
Now, if someone attacks you with a knife, LeiYunFat is right, you will almost certainly be cut. But that's not necessarily a big deal. There's a world of difference between a superficial slash (which would include something like a big ol' gash across the thigh) and having a part of your body impaled, or recieving a deep, focused, gruesome cut across your neck or something. Much of aikido is based on your attacker being comitted and focused enough to make them actually dangerous. If they're not, thwarting the attack is simple, even if they have a knife. Imagine some crazy idiot four year old girl coming at you swinging a knife around. Not all that scary, eh? In order to prompt a defense from you that requires more than a side step and an eye gouge, the attacker has got to be presenting a mortal danger to you.
Granted, this is much easier for them if they have a weapon. But the same basic concepts still apply. You're their target - probably a specific part of your body. Don't be occupying the space where the power of their attack is when the attack gets there. Redirect the energy that they had to use to penetrate the perimiter of your inate defenses and keep them off balance. No balance, no power, very little danger. If they're armed, disarm or destroy them. Don't mess around with that.
All that having been said, the inescapable truth of fighting is that someone willing to sacrifice their body for just a CHANCE to hurt you is very dangerous, deadly if they have a knife. There's no defense against being stabbed completely by surprise. Fear anybody with a gun. You can't dodge bullets, you know.