Adiabatic demagnetization and remagnetization in the rotating frame (ADRF and ARRF) are shown to be practical and efficient techniques for exciting and observing quadrupolar order, T-2,T-0, in NMR of quadrupolar nuclei such as H-2 or Na-23. A detailed theoretical description of ADRF and ARRF of spin I=1 and 3/2 nuclei. based on the well-known vector model of NMR, is presented and used to derive analytically a variety of pulse shapes for ADRF and ARRF. This theory is also used to calculate the state of the spin I=1 and 3/2 density operators following an ADRF or ARRF pulse and it is shown that the desired coherence transfer processes have the maximum amplitudes allowed by the well-known "universal bound" theorem. In principle, therefore, ADRF is shown to be superior as a method of exciting quadrupolar order to the Jeener-Broekaert experiment since the latter fails to excite the maximum T-2,T-0 amplitude for spin I=3/2 nuclei. The performance of ADRF is investigated using computer calculations and simulations and the conditions under which it yields broadband (i.e., non-oscillatory) excitation of quadrupolar order are derived. Using both H-2 (I=1) and Na-23 (I=3/2) NMR of liquid crystalline and biological samples, ADRF and ARRF rut demonstrated experimentally. The predicted broadband excitation behavior is observed far both ADRF and ARRF, whereas both the Jeener-Broekaert and double-quantum filtration experiments show excitation profiles that oscillate sinusoidally as a function of either the quadrupolar splitting parameter, omega(Q), or the duration of the pulse sequence, tau. Finally, a more general discussion of ADRF and ARRF of quadrupolar nuclei is presented and it is shown that the maximum coherence transfer amplitudes are achieved for nuclei of any spin quantum number I.