Quality racing game?

I'm looking for a good car racing game, that manages to achieve that feeling of speed as well as making the player sort of feel the grip on the track, so that it doesn't feel like it's floating or something. Simulation or arcade-ish, I don't care. Any recommendations? 

Need for Speed Shift 2 or the first Shift I thought were pretty good.

Depending on the type of game you want check out RFactor and GTR2 as well.

I highly recommend Need For Speed: Shift (the first game only, I thought Shift 2 was garbage) and Dirt 2.

Both of these games are a perfect blend of sense of speed, awesome graphics and great handling (once you become familiar with them).

Need For Speed Shift also sounds incredible on a good home theater system. Easily the best sounding racing game I have ever heard.

*Edited to add that I forgot to mention the F1 2010, 2011 and 2012 games. They are also very well done, and a lot of fun to play once you learn the tracks. Awesome rain effects, as well.

ACR is good and F2P (Of course with any F2P you can pay for stuff)

http://autoclubrevolution.com/en/

Dirt 2 is great too.

Dirt 2 > Dirt 3, loved it.

I used to play a lot of Live For Speed.

But note that the physics model is quite realistic and may feel "floaty" to some.

For arcade-style racers, I'd recommend the Trackmania games.

DIRT 2/Project CARS/DIRT3/GRID are all worth trying

Stay away from Test Drive Unlimited 2, not as good as the first one was.

It is the best precision driving game I've ever played, it also has one amazing and intuitivw track designer, great tracks, some insane;y fun roads, like a loop, or the wall slide in the trailer.
Highly recommend the game!
 

You should try test drive unlimited 2. I am having a lot of fun playing this game.

any racing title from codemaster is a safe option.

try race driver: grid. great game...made by codemasters(f1 2011, dirt) 

the new one is coming out at the end of the month. im rather excited for it. 

Need for Speed: Hot Pursuit relaunch on PS3, while not totally simulationlike, but also not as arcade like as for example the Underground Parts.

I still remember that awesoe feeling when you rushed through a tiny gap of two cars driving opposite direction with full power Nitro <3

In that moment you REALLY feel the grip ;)