Rather than do my typical analysis of the storyline of this, one of the most famous Batman stories of all time, I thought I'd focus instead on what goes wrong.
What goes right? First and foremost, the cover to Detective #31, which clearly belongs on any Bat-fan's top ten list:
What's new? The Batarang (here spelled Baterang) makes its first appearance, as does Batman's initial mode of aerial transportation: the Bat-Gyro. The story also features the first (two) death-traps for Batman.
What goes wrong? Lots; the story has more potholes than a New York street in late winter. The story starts out with Batman trailing Bruce Wayne's fiancee, Julie Madison as she (zombie-like) stalks a man:
What is she doing? Why is the man frightened of a young, attractive woman? Why is Batman following her? These stories are not really answered by the plot. Batman saves the man and then reacts in surprise when he realizes it's Julie, so apparently he was not trailing her, or at least not aware that it was his girlfriend. Julie comes out of her trance, and Batman takes her home, cautioning her to tell Bruce everything. She does so, and Bruce suggests that she see her doctor who tells her to take a cruise to Paris:
Okay, so the doctor is under the Monk's control as well? And despite obvious suspicions about his queeer behavior, Bruce sends Julie off on a luxury liner. He follows in the Bat-Gyro. briefly terrorizing the city:
That's one of the few times he smiles in the pre-Robin era; when he's panicked the citizenry. When he reaches the liner, he puts the Bat-gyro on autopilot and slips down to the ship, where he sees Julie. But before they can talk, the Monk makes his first appearance:
Batman manages to avoid being hypnotized by throwing the Baterang at the Monk, breaking the spell. But he then retreats back to the Bat-Gyro. Say what?
When the ship docks in Paris, Batman apparently misses Julie in the crowd of disembarking passengers, and thus must spend several nights trying to find her. When he does, he learns that she has a guard:
Are we to assume that she was able to sleep peacefully in a room with a giant ape? Batman ducks the monster, but falls into a net, which is then lowered by the Monk towards a snake pit. But his Baterang hits the lever to stop the downward motion of the net. It continues up and breaks a lightbulb, and Batman uses the broken glass to cut open the netting. Variations on that theme would prove to be a durable method of escaping death traps for Batman in the future.
The Monk escapes, but Batman rescues Julie and flies with her to Hungary, as the first part of the story ends.
In Detective #32, Batman is following a stagecoach. He overcomes the driver and throws a gas pellet into the passenger compartment. But the Monk isn't there; instead he finds a woman. Rather than apologize profusely for the mistake, he takes her back to his hotel and locks her in a room with Julie. Second blunder of the evening:
As you can probably guess, Dala is a vampire and has bitten Julie. Dala makes an offer:
But as they fly towards the Monk's castle, the Bat-Gyro is caught in a net. This time Batman is unable to overcome the vampire's hypnotic powers. The Monk summons Julie to join them; apparently she was within walking distance (never mind that Batman and Dala had to fly there).
The Monk informs Batman that Julie is destined to become a werewolf. Then he turns into a wolf himself and summons a pack, intending to have them kill Batman. Batman uses his gas pellets to kayo the wolves, but is unable to escape from the pit because his rope is too light to carry to a nearby post. Again and again he throws unsuccessfully, stopping every now and then to knock out the wolves with another pellet. Finally, after he runs out of gas, he hits on the bright idea of combining the Baterang with the rope and escapes the pit.
As it is dawn, the vampires are asleep and he does not hesitate:
He shoots the two vampires and flies back home with Julie.