Depends on the meat, depends on the cut.
Pigeon and duck - always rare. Absolutely nothing else is acceptable.
Venison - Extremely lean meat, should be rare as well, particularly if roasted.
Chicken/ pheasant - Not really a gradient here but overcooking ruins both of them.
Lamb - Again always rare, but that's out of choice.
Mutton - not sure why I'm including this but cook it long and slow.
Pork - similar to chicken, there's definitely a gradient here but it's much harder to see. I don't cook pork much.
With beef you really need to consider the cut when choosing how to cook it. The fat content determines how it should be cooked. For example, a fillet steak is extremely lean and should be eaten blue/rare for the best flavour. Overcook it and it looses all flavour, since the fat isn't there to provide it. Take a cheaper cut, such as flat iron, and the fat won't melt until you get to about medium-rare, so it'll just be chewy. Cook it to medium rare though and it's awesome. The highest fat content is in the stewing cuts. That's why you stew long and slow, often over 4-8 hours (use a slow cooker), to make sure all that fat melts and turns into flavour rather than chewy-ness.