While board games are generally seen as dedicated for at least three or four players, through history some exclusive 2-player games have appeared allowing a single pair of players to battle it out in private. In fact, the concept has become successful, promoting Duel versions of other games to eliminate the multiplayer feature or directly new versions with the gameplay adapted to just two players.
Lost Cities first came out in 1999. A small box containing a simple, elongated board where you could place the different cards as well as a deck of 48 cards, 12 of each color as we'll see ahead. Here's the display of the box and its contents.

The board shows "expeditions", which must be explored by playing cards. The deck includes three copies of a Handshake for each color, which serves as points boost, as well as a single card ranging from number 2 to number 10 in each color. Players draw a starting hand of 8 cards, placing the rest of the deck on one side away from the board, and the game can start.
Players have two actions available: they can mount an expedition, which involves playing a card on your side of the board under the appropriate color, or they can discard a card by placing it face-up on its colored space in the board. Whichever they do, they must then draw a new one so players always have 8 in their hand, and the twist is they can draw it from the deck, unknown to everyone, or draw the topmost card in any stack of the board, public knowledge. This is because the game ends when the deck runs out of cards to draw, after which the scores are tallied.
Expeditions must be mounted by playing cards in ascending order, first and foremost the Handshakes and then the numbers. You can start the expedition at any point, but keep in mind they can only go ascending from here. For example, playing a 3 means you can't play a 3 or a Handshake of the same color later, you must play a 4 or higher (no need for consecutive numbers, though this may have an advantage in the scoring). In the higher numbers this can mean trouble, because playing a 10 seals the expedition and you won't be able to play more cards there. Choosing the order is crucial, though most of the time you'll have to run risks because the cards needed are well down the deck or you may even draw the card you wanted right after playing the card you were saving but were forced to play.
The interesting part comes with the scoring. Each expedition you start means -20 points, and then you add to these the value of the cards you played. This means, you must play more than 20 points or else you'll score nothing or even negative points from an expedition!
Here's an example: you play a 4, a 8 and a 10 on the Yellow expedition. 4+8+10 means 21, so with the -20 for having started the expedition means you salvaged 2 points. Therefore, you must only start an expedition if you're quite confident you can at least compensate the opening deficit.
Handshakes offer a chance to raise the stakes. Each Handshake adds a multiplier to the score obtained above, meaning you can double, triple or even quadruple the obtained points. Following the example above, one Handshake would double the points and earn 4, two would earn 6 and all three would earn 8. But if your score is negative, they will also multiply it and increase the deficit further. In the other side, there's a 20 point bonus if you played at least eight cards in any single expedition, Handshakes included, to be added to the total score after the multipliers are applied.
Just if somebody is a bit lost with all the wording and such, let me show you the scoring through an image explaining it in detail.

The points of all five expeditions are added, and the first round is over. The scoring is carried over to the next round and then to the third and final round, after which the player with the highest score is the winner. Since the box doesn't include paper or pencils to keep the scores, it's quite easy and faster to play just single-round games. I've yet to play a three-round game in real life due to this, so far I've only done it online.
In 2018 the game got a revision and now has six suits of colors, but as far as I'm aware this sixth color is optional for gameplay meaning you can use just five as in the original version. I haven't yet seen this version, but since this game could be a nice addition to my stash I might look into it, and surely the new versions will be more available and better priced than the classics.