Personally, I love playing/running wizard characters. I've tried almost every base class and yet still find myself once again rolling up a wizard. The appeal to the wizard class for me is two-fold, one: I've wanted to be a wizard since I was five years old having the Hobbit read to me, and two: the marvelous diversity encompassed in the class.
Most DM's approach the subject of magic from their own perspective. DM's are left to build the arcane arts into a world where fireballs and teleportation are realities.
The books leave the subject of magic implementation up for interpretation, some DM's support the notion that anyone can be a wizard just by choice and that every lowly peasant has access to arcane comforts. And this is often a path taken by DM's. Making for a truly interesting game however requires some extra DM homework:
Dungeon Master's have to make demographic decisions about magic, is it commonplace in the world? If so who uses it? What are spellcasters like? What is magic's political role? Answering these questions adds a completely new dimension to the game and also gives spellcasters information to spice up their backstories.
Similarly a DM must decide the actual mechanics of magic. And I'm not just talking verbal and somatic components... Spellcasting makes a huge access point for role-playing. Throughout human history there have been many types of magic and spellcasting and it is up to the DM to decide if these other types of magics exist in the world, and if your PC's could potentially learn them.
A DM is free to decide just how magic works in the world, if it operates by a set of scientific principles or if it comes from a specific source. Some of my favorite games have involved scholar characters who can only perform magic through stringent adherence to rules, it adds a sense of responsibility to the act of casting magic. There are two specific rules traditionally associated with spellcasting that I like my players to incorporate into their game:
1. The Law of Contagion- Things once in contact continue to interact after separation, anything once in contact can be used as a "witness". Traditionally exemplified by the use of a persons hair in creation of a voodoo doll
2. The Law of Sympathy- The idea that if the cause and effect of an action resemble a different action, they have resonance with each other. For example, if you wish to cast a fly spell, you might jump around with feathers like a bird.
Adding rules or systems like these to make scientific principles for magic in your game adds a depth and difficulty to playing arcane classes that the true role-players will definitely appreciate.
I've played under multiple systems of magic (some completely designed from scratch) but traditionally, the more elaborate they are, the happier a wizard I am :)