Palworld

https://palworld.th.gl/
Credit to embarrassed-notice32 on Reddit.

This interactive map is a very useful tool for finding the Pals you are yet to catch. I believe there are 111 (113 maybe) Pals in the game currently and for this achievement you will be required to catch 90.

Extra tips
- increase your catch rate in custom settings (although I’m not sure if this does anything)
- use better pal spheres for the harder to catch Pals
- offer lifmunk effigies to the statue of power to permanently increase your catch rate
- remember some Pals will only appear at night
- Elemental variations do count
- as per Healtti’s comment, flying around and looking for eggs is a really useful way to get some of the harder eggs, you’ll need an incubator at your base. The volcano and ice area gave me some good eggs.
- Tainted Sparta commented that if you catch a Pal Merchant or black market trader you can put them down at your base and buy Pals from them. Quit and reload to change the selection on offer (I didn’t do this but could be a good way to get the last few you need if you have lots of coin)

A link below with more detailed information:
https://videogames.si.com/guides/palworld-increase-catch-rat...

Edit: this list isn’t exhaustive. There are some pals that aren’t on there yet as it’s a work in progress. But it certainly helps on your way to 90

Edit 2: just to confirm, you must be in your own world to unlock achievements. This applies for all Palworld achievements currently.

Edit 3: added some useful user comments to the solution for ease of access.

(Feb 23 - reports that since a game update you should be able to make progress to this playing in other peoples worlds

If you're having trouble capturing higher level Pals and don't feel like going through the hassle of fabricating high-end Spheres, one really solid option is to straight up buy them. 

There are regular Pal merchants in every town, and some hidden Black Market merchants.
You can check their location here:
https://mapgenie.io/palworld/maps/palpagos-islands

Some merchants are specialized, e. g. they only sell sand-related Pals, or only fire pals. You can tell from where they're located.

Buying Pals got me from around 70 to the desired 90 with very little trouble.

If you're low on money, one very easy way to become rich is to farm Precious Pelt, Plumes and Entrails from bosses. You can trivially defeat them by tinkering with the settings, as explained in many other guides.

I hope this helps!

EDIT: Oh, and if you're wondering, different "versions" of the same Pal count. E.g. Mau and Mau Cryst count separately.

EDIT 2: You can reset merchants' inventories by fast-traveling or by just quitting and reloading the world.