A mean-square convergence theorem is proved for strong numerical approximations of stochastic differential equations whose coefficients satisfy a one-sided Lipschitz condition and may grow polynomially at infinity. Implications regarding almost sure convergence are given. An (explicit) balanced method is proposed and is proved to have strong convergence with order \({1\over 2}\). Fully implicit methods are analyzed with special attention devoted to the midpoint method. Results of numerical experiments using seven different schemes on two examples are discussed with respect to accuracy and computational costs and show agreement with the theory presented.