I am interested in using a diceless resolution system for my roleplaying, but am not sure what my options are in that vein. I can think of basically two categories of diceless tabletop role-playing systems - systems which eschew any randomisation, and those which randomise by some method other than dice.
Non-random tabletop systems
One option is of direct comparison of stats or other game statistics without a randomization factor. The obvious example here is Amber DRPG, but I suspect that there are other narrative driven RPGs out there that I don't know about.
Even in Amber, situation resolution it is rarely by direct comparison between stats. Narrative techniques can be used to manoeuvre two characters into a situation which narrows the gap between their ranking, widens it, or brings other stats into the picture - the classic example of course being Corwin relying on his stamina to stretch out conflicts with his siblings such that it becomes a dominant enough factor that he can escape.
Other randomisation techniques
I have seen card-based randomization system. Over the years, I've played quite a few home brew games based on the original SAGA Dragonlance system, which was similar to the SAGA Marvel superhero system.
The basic mechanics were that you had a hand size according to your experience, and for any skill check you would play a card from your hand and replace your card from the draw pile. If you had a card of the right suit as the skill check (a strength card on a strength check) you would also draw a card from the draw pile and add that to your skill check, so a small trump was often worth more than a medium value card not of the correct suit. When you were wounded, you would discard cards to the value of the damage, but those cards wouldn't be replenished - your hand size would be reduced until you were healed. One nice aspect of the system was that it simulated fatigue beautifully, you might start a scene with a spread of high, medium and low cards, but as your high cards were played they would more often than not get replaced by lower value cards, until you were left with a bunch of useless cards and feeling very vulnerable.
The only other system I've played without dice was Everway which, if I remember correctly, used a combination of these techniques, stats based resolution, card resolution and GM fiat.
What I would be interested in is what other diceless resolution techniques exist, ideally with examples of tabletop diceless resolution methods people have tried and how they found them to play.