We establish some general theorems for the existence and nonexistence of ground state solutions of a steady-state N coupled nonlinear Schrodinger equations. The sign of coupling constants is crucial for the existence of ground state solutions. When all coupling constants are positive and the associated matrix is positively definite, there exists a ground state solution which is radially symmetric. However, if all coupling constants are negative, or one of them is negative and the associated matrix is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N = 3.