It lived for a while but never really grew up. The Bethesda Launcher arrived in 2016, but it unfortunately didn't evolve into much of anything besides a desktop client with two buttons: buy game and play game.
And now it's being switched off. On May 11 the Bethesda Launcher will become officially defunct, and as promised, starting today you can migrate your Bethesda Launcher games library over to Steam. Note: You'll still be able migrate your games after May 11 when the launcher is dead. You just won't be able to use the launcher itself anymore. So, you might as well do it now.
Bethesda has a full list of instructions on how to begin the great Steam migration, but basically you'll start by trying to log on to the Bethesda site, realizing you have no idea what your password is because you haven't logged in there in ages, resetting it, and then logging into the Bethesda site. On your account page, there are sidebar tabs to link your Steam and Bethesda accounts (if you haven't already) and to begin the transfer process.
I did it this morning and it went without a hitch, though your experience may differ: I only owned 4 games through the launcher and I'd already linked my Steam and Bethesda accounts for some long-forgotten reason. But, it worked and only took a few minutes. My Bethesda games are now safely in my Steam account, and it even transferred the Fallout 76 Atoms I had in my Bethesda wallet.
Transferring your saved games may be a bit more of a hassle, depending on which games the saves are for. So let me direct you to the FAQ on transferring save progress, where you can check each individual game to see if any extra effort is required on your part. The migration will bring over save files automatically for a number of games (like Fallout 76, Fallout 3, and Doom 3) but for others like Deathloop, you'll have to manually copy the data from the original location and plop it into the new one.
Transferring save data for Doom Eternal seems especially complicated because it won't transfer automatically and can't be done manually. It actually needs to be done with a conversion tool. Doom Eternal has its own separate list of instructions here. Frankly, it sounds like hell.
As for subscriptions to Fallout 76's Fallout 1st membership, that's got a whole page of its own. An existing membership will continue until it naturally expires, but renewing an expired membership or starting a new one will need to be done through Steam from now on. Though the Bethesda launcher will continue to work until May 11, all real-money transactions are halting now.
The data migration hasn't been smooth sailing for everyone, and some players have registered complaints about the account linking not working. Other issues might be resolved by installing and launching a game once it's migrated to Steam: for instance, some users reported purchasing the deluxe edition of Rage 2 from Bethesda but found just the standard edition had migrated to their Steam accounts. Apparently, once the game is actually downloaded and launched on Steam, the Deluxe DLC will appear.
If you're struggling, you can always create a support ticket here, and then cross as many fingers as you can that you'll receive a solution quickly.