Gin Rummy (and many variants of Rummy) was common here in U.S. My favorite to play was so-called "Contract
Rummy". Works similar to 'plain' Rummy except as follows;
1) there are a set number of rounds, the version I played always had 5.
2) each hand has a specific assortment of hands you need to make before you can discard, to go out (get rid of all cards, thus ending the round). 2 sets or 2 runs or 1 run and 1 set etc. I can't remember the exact rules for each round anymore except that it got more difficult as the game progressed.
3) I don't remember if this was singular to this particular game or not but a player could "buy" a (1) card from another player by taking another card from the pile, but only 1 per turn or something like that.
4) Every round played like normal rummy (players lay their cards down whenever and first person to throw their last card away ends the round). But the final round functioned like Gin Rummy; in order to lay out you have to have all the cards in your hand fit the contract, thus the first person to lay out also ends the round and everyone else is stuck with all the cards in their hand.
The rest I think functions mostly like Rummy. After the final round the player with most points wins.
There's also this game called Rummikub; which I never played but looks like a mix between Rummy and dominoes... no idea how it works.
https://en.wikipedia.org/wiki/Rummikubhere's another Yatzhee like game I've not really played but heard of:
https://en.wikipedia.org/wiki/Farkle