Before I say anything about the content, let me just say that I personally love this series. I think the characters have great depth and I love how Clare continues to develop them throughout the series. Now content wise, parents and teens should be aware of a few things before you dive right in. There are several couples, but so far Jace and Clary have gone farther than anyone else. In the heaviest scene involving the two, Clary strips to her bra and underwear and Jace is shirtless, but they never go any further. Other couples, including a gay couple, kiss once or twice. There is some language, but nothing too bad. Keep in mind that these books are centered around teens, and most teens do cuss, especially in tense situations. There are several violent scenes that border on disturbing *mild spoiler alert*, such as a room full of mutant, dead babies, and in one scene a vampire is forced to drink the blood of a (presumably) dead human, which might make some readers queasy. There are several mentions of throats being slit, heads being cut off, people dismembered, vampire feedings, etc. To parents concerned about the supernatural/demon aspect of it, it's nothing that (in my opinion) would scare or scar a moderately mature teen.
Personally, I don't think that any of the subject matter is too heavy for moderately mature teens, but I wouldn't (personally) let my kids younger than 13 read it just for some of the the heavier subject matter.