1. The covariance of two constants, c and k, is zero.
   $Cov(c,k) = E[(c-E(c))(k-E(k)] = E[(0)(0)] = 0$
2. The covariance of two independent random variables is zero.
   $Cov(X, Y) = 0$ When X and Y are independent.
3. The covariance is a combinative as is obvious from the definition.
   $Cov(X, Y) = Cov(Y, X)$
4. The covariance of a random variable with a constant is zero.
   $Cov(X, c) = 0 $
5. Adding a constant to either or both random variables does not change their covariances.
   $Cov(X+c, Y+k) = Cov(X, Y)$
6. Multiplying a random variable by a constant multiplies the covariance by that constant.
   $Cov(cX, kY) = c*k \: Cov(X, Y)$
7. The additive law of covariance holds that the covariance of a random variable with a sum of random variables is just the sum of the covariances with each of the random variables.
   $Cov(X+Y, Z) = Cov(X, Z) + Cov(Y, Z)$
8. The covariance of a variable with itself is the variance of the random variable.
   $Cov(X, X) = Var(X) $

For additional info see http://prob140.org/textbook/content/Chapter_13/02_Properties_of_Covariance.html