Dwarf, elf, hobbit, and man; these races were central to the Fellowship of the Ring in the trilogy by J.R.R. Tolkien. You could choose to play the same race as Gimli (dwarves), Legolas (elves), Aragorn (humans), or Frodo (hobbits, though technically Dungeons & Dragons calls them halflings).
Around the same time, in AD&D 2nd edition you could also find the gnome and half-elf races in the Player's Handbook. [ed. note: They had also been in the 1st edition release but 2nd edition failed to carry forward the half-orc.] The next edition threw the half-orc race back into the mix of starting options (it had been removed between 1st and 2nd edition AD&D). The v.3.5 edition did not add any new races to the Player's Handbook but races and subraces had been added to D&D in supplement books. The 4th edition Player's Handbook brought a supplemental race (tieflings) into the core, added a reptilian race (dragonborns), split elves in twain (forming elves and eladrins as distinct races), and removed two options (gnomes and half-orcs).
In his regular column, Chris Perkins wonders "how many race options a campaign (not to mention the game) really needs". You can even take a couple polls on the topic. I prefer worlds dominated by humans with dwarves, elves, and other races in secondary roles. I'm no slouch at playing demihumans, I've played many dwarves and even a dragonborn, yet I want prominent humans in the adventuring party of player characters. My groups have tried different experiments with character race.
Ed Matuskey. The setting featured a pseudo-European map. Russian geography and Russian culture defined elves. Halflings were the French. German city-states were made up of dragonborn. England was an intermixing of eladrin (fey) and humans. The Scandinavian region was populated by dwarven pirates (vikings).
There are gameplay problems that arise when one race defines a culture and dominates a geography. The player characters stick out like a freak show if they're in elven Russia when the party consists of a, relatively normal, mix of three humans, a dragonborn, and a halfling. It's immediately obvious to the NPC world you're where you don't belong if no one of non-elven descent should be there. Luke Skywalker and Han Solo were able to disguise Chewbacca as a prisoner going to the detention cells because it was believable that the two humans were stormtroopers. A magical hat of disguise can solve this problem in D&D but in a low magic game, which I prefer, that is an unpalatable solution. Race can be a barrier to subterfuge if the campaign setting is set up in certain ways.
We've also gone to the other end of the spectrum. In another campaign, all characters looked human with slight racial variations. The human character that used half-orc rules had attribute bonuses as if they were a half-orc but she looked like a human with green skin (anime style). Mix in the diversity of mankind and deception was easy. People of all these Dungeons & Dragons campaign nations mingled and NPCs didn't usually give a second thought to superficial differences in appearance. This was a great setting but it did lack in the traditional fantastical feel of the genre (on purpose, mind you). Just be cautious, and know what you're getting yourself into.
So what is the proper role for race and the proper options? Monte Cook ran a poll on the issue of Racial Importance. Players seem to want a variety of races and they want them to be relatively simple. How should the game handle hybrid races (those pesky half-elves and half-orcs). Go ahead an vote for which D&D races you have played.
Characters races and their impact on the game is a large topic. I'd like to focus on ability score bonuses. When a +2 Strength comes with very few races and is key to being a fighter, it is hard to justify a racial theme as "often warriors" without that bonus. Dwarves were supposed to be martial warriors but they get a bonus to Constitution so half-orcs, as a race, are better fighters than dwarves. The Essentials product (D&D 4e) let dwarves swap their +2 Wisdom for +2 Strength instead. I assume this is so that dwarves would make mechanical sense as iconic fighters. It definitely made already powerful dwarves the master defenders. I think this needs a solution.
The simplest solution is expunging racial attribute bonuses from the game. I think this is unlikely to be adopted by Wizards of the Coast. Except for Basic Set rules, races have always had their attribute modifiers. To keep the flavor of the races (tough dwarves, dexterous halflings, &c), you implement the Basic Set rule of minimum ability scores for each race: no one in the race is "bad" at their thematic good stat but the class appropriate attribute needs are the same across all player character races.
If D&D Next uses dice to determine ability scores, you could keep the +1 or +2 attribute bonuses for each race if you instituted a cap of 18 on all stats. Sure, if a +2 Constitution dwarf rolls a 17 and puts it into his Constitution he'll only have an 18 Constitution but he could also put that 17 into his Strength if he so chooses. In this solution an 18 represents the maximum physical potential of your attributes and races don't violate that physical barrier.
In a point-buy system you can charge demihumans an appropriate number of points if they want to have a superhuman 20 Strength (only half-orcs need apply). Instead of getting a +2 Strength, whether you're a wizard or a fighter, everyone pays the same number of points for an 18 Strength and the half-orc has the option to spend additional points to reach 19 or even 20. This avoids the 3rd edition problem where one stat bonus (+2 Strength) was considered better than others (+2 Constitution) so that the half-orc had to have a -2 penalty to Intelligence and to Charisma to balance out his more powerful +2 because his +2 applied to Strength instead of applying to some "lesser" attribute.
Lastly, another fine option is to limit the scaling of attribute based modifiers. An 18 doesn't need to offer a 33% better bonus to attacks (and damage rolls) than a 16. Under 2nd edition, there was no difference in bonuses on some stats between a 16 and a 17 or between a 17 and an 18. Flatten the power curve.