To approximate solutions of stochastic differential equations (SDEs), explicit Euler-Maruyama (EM) schemes have been used most frequently under global Lipschitz conditions for both drift and diffusion coefficients, whereas implicit schemes have been used widely for SDEs without this condition but require additional computational effort for the implementation. In addition, tamed EM schemes and the truncated EM schemes have recently been developed for SDEs without satisfying the global Lipschitz condition. Taking advantages of being explicit and easily implementable, modified and truncated EM schemes are proposed in this paper. It is shown that our modified truncated EM schemes preserve the asymptotic pth moment boundedness of the underlying SDEs. Furthermore, different schemes are constructed to approximate the dynamical behaviors such as the exponential stability in pth moment and stability in distribution. Several examples are given to illustrate our findings.