One can infer the orbital alignment of exoplanets with respect to the spin of their host stars using the Rossiter-McLaughlin effect, thereby giving us the chance to test planet formation and migration theories and improve our understanding of the currently observed population. We analyzed archival HARPS and HARPS-N spectroscopic transit time series of six gas giant exoplanets on short orbits, namely WASP-77 Ab, WASP-101b, WASP-103b, WASP-105b, WASP-120b, and WASP-131b. We find a moderately misaligned orbit for WASP-101b (lambda = 34 degrees +/- 3) and a highly misaligned orbit for WASP-131b (lambda = 161 degrees +/- 5), while the four remaining exoplanets appear to be aligned: WASP-77 Ab (lambda = -8 degrees(+19)(-18)), WASP-103b (lambda = -2 degrees(+35)(-36)), WASP-105b (lambda = -14 degrees(+28)(-24)), and WASP-120b (lambda = -2 degrees +/- 4). For WASP-77 Ab, we are able to infer its true orbital obliquity (Psi = 48 degrees(+22)(-21)). We additionally performed transmission spectroscopy of the targets in search of strong atomic absorbers in the exoatmospheres, but were unable to detect any features, most likely due to the presence of high-altitude clouds or Rayleigh scattering muting the strength of the features. Finally, we comment on future perspectives on studying these planets with upcoming space missions to investigate their evolution and migration histories.