Today’s smartphones are no more normal mobile phones, they are small personal computers. Therefore threats for smartphones are the same as for personal computers. The only difference form this point of view is that personal computers generally use Microsoft Windows as operating system, whereas there are a wide variety of mobile platforms. Due to this fact mobile platforms are harder to attack.
But the biggest danger for a mobile phone is one that is not one for personal computers: You can lose your mobile phone. This is not only a danger for smartphones but also for all mobile devices containing personal information. Another danger for smartphones is leaving them unattended or loaning it to people because they can install spyware on your smartphone. Additionally to spyware there are mobile viruses, worms and Trojans threatening your smartphone. They can spread using e-mail or via SMS.
Generally speaking, web-based and e-mail attacks are as possible with smarthones as they are with personal computers. Wi-Fi and Bluetooth are two technologies which can be dangerous too. Wi-Fi can be attacked by a man-in-the-middle attack and Bluetooth is also a target for attacks. A good discussion of Bluetooth security you can find here: part1, part2. Here you can find a FAQ on how to use your smartphone securely. Another interesting article about smartphone security can be found here.