What can I use as a substitute for eggs in a vegan recipe?

I've gone vegan, and I'd like to bake some of my favorite cakes and cookies, but the majority of the recipes call for eggs. What can I use to replace the eggs? I've heard that I can just leave them out, but that I may end up with a cake that is on the dry side. Help!
Thank you!

How strict of a vegan are you? Because if you can't eat eggs, I would assume, you wouldn't eat butter, milk etc. Found this website that has other things that can sub eggs:
Ener-G Egg Replacer - follow directions on box.
2 tbsp corn starch = 1 egg
2 tbsp arrowroot flour = 1 egg
2 tbsp potato starch = 1 egg
1 heaping tbsp soy powder + 2 tbsp water = 1 egg
1 tbsp soy milk powder + 1 tbsp cornstarch + 2 tbsp water = 1 egg.
1 banana = 1 egg in cakes.

2 things I have found that work.
EnerG Egg Replacer - found in health food store & health food sections at grocery store.
&
When baking, one of the best egg replacers is to mix 1/4 cup ground golden flaxseed meal in 1/4 cup boiling water for each 1 to 2 eggs called for in recipe. The flax actually helps your baked goods to rise. This works especially well with cakes and breads. Another substitution that works with crisp cookies is blending 1 teaspoon baking
powder with 1 Tbsp. liquid and 1 Tbsp. vinegar.

Slaint茅(to your health)

use Ener-G Egg Replacer and for butter u can use Willow Run brand soy butter and for milk you can use soy milk. i sugest you use almond milk instead of soy. (better nutrients and all that good stuff) and it tastes just as good. try to stay a way from rice milk.

Ener G egg replacer or Flax seed or even tofu

i have vegan cookbooks
depending on the recipe, i have seen where applesauce is substituted
also you can buy egg replacer - it usually works fine

Haven't tried for cakes yet, but I like 2Tbsp ground flax seeds+ 3 Tbsp water (which equals 1 egg), mixed and then let sit for a minute or two, in my muffins, cornbread, waffles and pancakes. It makes them moist and helps it rise a bit! Not to mention the extra goodness of the flax seed!

Applesauce, Tofu, and Egg Replacers work well, however I use my own mixture by putting in Soy Milk, and if it gets to liquidy, I add some more Flour, and a pinch more of Brown Sugar....

It works pretty well for me, no one ever figures out that they are vegan, until they see me eat them...