Completing Hearthstone's Adventures on Heroic is meant to be a challenge taken only by the toughest and most patient of heroes. But take heart, fellow Explorers. With a little deck editing, you can take them all down, earning your first step towards a special Heroic card back and all the bragging rights that come with it:

We've wrestled with the intense challenges of The League of Explorers on Heroic Mode, and outlined our winning strategies below. These strategies should largely work with the Normal bosses as well.

This week: The Ruined City.

Slitherspear 

Enraged!: Give your hero +2 attack this turn.
Getting Hungry: Summon a series of Nagas. (See below)
Endless Hunger: Summon a Hungry Naga.

Slitherspear is the toughest boss in this wing's set, summoning an increasingly powerful Naga every turn alongside some regular minions. It's very easy to get overwhelmed, and the longer the match goes, the more insurmountably beefy the minions will be. 

As you may expect from such an overpowered boss, the Freeze Mage is your best bet. Though you may also want to attempt a Handlock, which uses lost health to its advantage, a Freeze Mage is more predictable. The trick is to prolongue the match as long as you can with freezing spells and Doomsayer. If you can keep two Sorcerer's Apprentices on the board (Echo of Medivh may help), your Archmage Antonidas will be able to fire off a a whopping 5 Fireball spells per turn, for a total of 30 damage. That should be more than enough to clear the board and damage Slitherspear to boot. Even using this strategy, though, it can be tough going and may require a few tries to get the right draw order.

Recommended Class: Mage
Recommended Cards: Freezing Spells, Doomsayer, Sorcerer's Apprentice, Archmage Antonidas, Reno Jackson

Giantfin

Mrglmrgl MRGL!: (0 mana) Draw cards until you have as many in hand as your opponent.
Heroic: (0 mana) Draw two cards.

On Normal mode, Giantfin's Hero Power makes him match your card draw while he floods the board with Murlocs. This makes an aggressive "Face Hunter" deck ideal, but the slight change to his Hero Power in Heroic makes him considerably harder. Still, Hunter is the best choice, albeit with some slightly more control-oriented tools. Hunter has lots of ways to deal small amounts of ping damage to lots of enemy minions. Given that most of Giantfin's Murlocs have just one or two health, all you need to control the board is lots of AoE damage options.

Knife Juggler is a great pick, especially when paired with Snake Trap and Unleash the Hounds to get double the value from your one-damage pings. Explosive Trap is perfect for clearing the board in a pinch, since most of the Murlocs max out at two health. Unstable Ghoul and Dreadscale are solid ways to clear the board too. Hungry Crab is a good way to develop a relatively big minion while also serving as instant Murloc removal for bigger threats. Once you've established board control, just keep taking out his Murlocs as they come and hit him with whatever remains. 

Recommended Class: Hunter
Recommended Cards: Knife Juggler, Snape Trap, Explosive Trap, Unleash the Hounds, Hungry Crab, Dreadscale

Lady Naz'jar 

Pearl of the Tides: At the end of your turn, replace all minions with new ones that cost (1) more.
Heroic: At the end of your turn, replace all minions with new ones. Yours cost (1) more.

Lady Naz'jar's power assures that she'll always be outpacing you for tempo. In fact, one of her first moves is usually to drop Mirror Image and then immediately swap it for two 1/1s. However, two can play at that game. Even though your minions won't be getting an upgrade, you can still utilize the same basic strategy to flood the board and give her too many small threats to deal with. Keep in mind that it doesn't matter if your minions are hurt, just if they survive to the changeover.

Pack your deck with any card that produces multiple bodies. As a bonus, most minions that produce more than one body have underpowered stats, so even being swapped for a minion of the same cost will usually be an upgrade. 

Murloc Tidehunter, Echoing Ooze, and Imp Master are all good choices. Keep in mind that Deathrattle effects probably won't trigger, so your main priority should be a sheer mass of minions. For minions that produce another body as a Battlecry, pack Brann Bronzebeard to double the effect. Shaman is a natural choice for this, since Feral Spirit gives you two 2-mana minions. Focus on clearing out the big threats first, and then keep flooding the board and taking out the smaller ones as she runs out of steam. You can pick away at her health slowly, or pack a Bloodlust or two for a big finisher.

Recommended Class: Shaman
Recommended Cards: Feral Spirit, Murloc Tidehunter, Echoing Ooze, Imp Master