The latest single-player adventure for Hearthstone, "One Night in Karazhan", has just recently released its final section.
The theme of "One Night in Karazhan" is a party thrown by the wizard Medivh, which isn't going quite according to plan. A demon lord is crashing the party, and pulls Medivh through a portal into his domain when the wizard tries to evict him. (That's the very first fight, which is available to anyone to play for free.) You have to make your way through the wizard's tower to the top, then rescue Medivh. It's fairly silly, but that's the idea with Hearthstone stories - light and fun, with plenty of humor to keep you entertained through the various battles.
After that intro fight, you have to purchase each of the four sections, either via a microtransaction or using in-game gold. I used gold since I have plenty from all the daily quests that I've done. Blizzard releases the sections one week at a time, but once they're all out then you can play through the whole thing at whatever pace you like. You get plenty for your money - the adventure itself, the class-specific challenges, and quite a few cards.
If you're really feeling adventurous, you can also try all the fights on heroic mode, which gives the bosses huge advantages that are very difficult to overcome. I've done some of those in the past, but not this one yet.
I had fun with "One Night in Karazhan", and I'd recommend it to anyone who enjoyed Hearthstone enough to play it regularly.