Monotonicity criteria are established for the generalized Marcum Q -function, QM(alpha,beta), the standard Nuttall Q-function, QM,N(alpha,beta) , and the normalized Nuttall Q-function, QM,N(alpha,beta), with respect to their real order indices M,N. Besides, closed-form expressions are derived for the computation of the standard and normalized Nuttall Q-functions for the case when M,N are odd multiples of 0.5 and M ges N. By exploiting these results, novel upper and lower bounds for QM,N(alpha,beta) and QM,N(alpha,beta) are proposed. Furthermore, specific tight upper and lower bounds for QM(alpha,beta), previously reported in the literature, are extended for real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, in the information-theoretic analysis of multiple-input multiple-output systems and in the description of stochastic processes in probability theory, among others."This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder."