Ars Technica had a 'best two player' guide a while back:

https://arstechnica.com/gaming/2016/04/table-for-two-our-fav...

My partner and I are always on the lookout for good two player games to fit into the gaps. Here are a few:

Morels - A game about gathering mushrooms and cooking them for points. The game simulates a walk through the forest with a rotating track of decaying mushroom cards. A really pleasant game for a Sunday morning.

Trambahn - Building tram tracks in Germany (Austria?) from horse-drawn buggies to more modern electric trams. Completely card based and has some clever mechanics re: scoring and building stations.

The Bloody Inn - You and a partner(s) are running an inn with the sole purpose of amassing money, usually by robbing/murdering people in their sleep. Fun game that can actually play up to 4 but works well with 2.

Pathfinder Adventure Card Game & Arkham Horror Card Game - These are both far heavier than the rest, but are great if you're into D&Dish adventure games but don't have a group for D&D. You essentially work through campaigns and scenarios with a character by playing again the deck/board, so no DM necessary. We like both of these a lot.

Race for the Galaxy is fantastic for two players; my partner and I have probably played it a hundred times or more...

Elder Sign is generally great, but collapses down to two players nicely.

Seven wonders duel is a great game, but can be a bit too zero-sum; we tend to prefer games where we do well on our own terms, without needing to block or disrupt the other's game to get ahead...

Carcassonne is fantastically simple but still very deep/complex strategically. Play without Farmers for a few games before adding them and decide whether you want to keep them or not - they make move planning much more abstract which is both good and bad. My friends and I play with and without them regularly, usually depending on how many beers we've had :).

Seriously though, Carcassonne has taken over my gaming group. We still play other games now and then but we've played it more than anything else by a long shot, and we regularly play with 2 all the way through the max of 5 people and it's fun in all incarnations. Truly incredible for a game of its simplicity.

Carcassonne can be surprisingly tense in 1v1 and expands nicely to accommodate as many players as you like. There are very, very, very few games that achieve this.

If you two like Chess but want something a bit more zany and light, I recommend The Duke. You each start out with nothing but a "king" (duke) and he has the power to let you draw random pieces out of the bag, and each piece has unique movements. Actually, two sets of unique movements, you flip the piece over after it moves.

The cool part? The pieces are self-documenting: they're flat tiles with their moves printed right on them.

That Ars article is a good start. Some games me and my girlfriend particularly enjoy

* 7 Wonder Duel - A specific 2 player only version of 7 wonders * Jaipur * Stronghold

My wife and I love pass-and-play with the Talisman phone app, the actual board game would be too much effort.

You mustn't treat it like a serious strategic chess-like game, but the emergent stories, triumphs and disasters are good enough to keep us coming back. I've bought all the expansions.