The pulsed-field-gradient spin-echo NMR measurements have been performed on 1-ethyl-3-methylimidazolium bis(fluorosulfonyl)imide ([emim][FSI]) and 1-ethyl-3-methylimidazolium [bis(trifluoromethy)sulfony]imide] ([emim][TFSI]) over a wide temperature range from 233 to 400 K. Molecular dynamics (MD) simulations have been performed on [emim][FSH], [emim][TFSI], [N-methyl-N-propylpyrrolidinium][FSI] ([pyr(13)][FSI]), and [pyr(13)][TFSI] utilizing a many-body polarizable force field. An excellent agreement between the ion self-diffusion coefficients from MD simulations and pfg-NMR experiments has been observed for [emim][FSI] and [emim][TFSI] ILs. The structure factor of [pyr(13)][FSI], [pyr(14)][TFSI], and [emim][TFSI] agreed well with the previously reported. X-ray diffraction data performed by Umebayashi group. Ion packing in the liquid state is compared with packing in the corresponding ionic crystal. Faster transport found in the FSI-based ILs compared to that in TFSI-based ILs is associated with the smaller size of FSI- anion and lower cation-anion binding energies.:A significant artificial increase of the barriers (by 3 kcal/mol) for the FSI- anion conformational transitions did not result in slowing down of ion transport, indicating that the ion dynamics is insensitive to the FSI- anion torsional energetic, while the same increase of the TFSI-anion barriers in [emim][TFSI] and [pyr(13)][TFSI] ILs resulted in slowing down of the cation and anion transport by 40-50%. Details of ion rotational and translational motion, coupling of the rotational and translational relaxation are also discussed.