Amber is a classic example that comes up in these discussions.
Intrepid uses playing cards instead of dice to resolve scenes and combat. For scenes, two people each narrate an outcome, and players vote on the version they prefer using red or black playing cards. A card is then selected and that outcome becomes the truth, so there is still an element of randomness.
In combat, each suit has a specific meaning for the ebb and flow of the battle, jokers change the scene, and the first person to draw the 4th ace wins the fight.
Most of Ben Robbins' don't have a random element at all and conflicts are resolved through procedure. Follow uses two coloured stones/poker chips/tokenswhich are drawn from a bag, similar to Intrepid. He also provides a "finger dice" system for getting dice-like results without using dice. On a signal, every player throws from 0 to 5 fingers, and groups of 5 fingers are eliminated. The remainder is roughly equivalent to a d6 roll.
Grab a deck of cards. Hold it face down, so the backs are on top. Take the bottom card and flip it so its back is on the bottom. Now the deck should have a back on both sides.
Go up to someone, ask them to pick a card at random. Fan out the deck, making sure not to reveal the flipped bottom card.
Ask them to study the card carefully and memorise it. While they are looking at it, flip the deck so the bottom card is now on top. The deck looks the same to them, but now all the cards other than the new top card (the bottom card you flipped previously) are now face up.
Take their card and slide it into the middle of the deck. (don't fan out again, obviously, that will spoil the trick).
Announce you will find their card in the deck. Place it behind your back. Quickly do two things: flip the top card around so it is now facing the same way as the rest of the deck. Then rotate the deck once more so all the cards are facing down, backs on top.
Bonus points if you then make some obvious flipping/shuffling sounds.
Bring the deck back in front of you, flick the top with your fingers, and then fan it all out. The card they picked will be the only face up card in the deck.
Works best for just one person you are trying to impress, too easy for gawkers to notice you rotating the deck. Though if you are better at sleight of hand than me you could probably get away with it by asking your volunteer to make a big show of showing everyone the cars so they can all see.