Does normalizing affect quality?

[GS1984]

Can somebody please explain to me whether audio normalizing affects quality?

I have some music files in wave format which are way too low. The peak is at -12 db or so. If I normalize the files to somewhere around -3 or -1, does that worsen the quality in any way?

And here's another scenario:
I had some music wave files. They were too loud (peak at 0db) and I didn't like that before, so I normalized them down to -6db and saved them like that. If I now want to normalize them back to around -1 db peak, will the going back and forth affect the quality?

I'm quite confused on this stuff. Any help will be appreciated. Thank you.

BTW I'm using Adobe Audition.

[DFKT]

Every step affects/lessens the quality of the audio signal.

I only know of one non-destructive method of normalizing: it's MP3Gain, but only works on MP3s...

[Axed]

Replaygain is also another "lossless" way to sort of normalise audio. If your dealing with music that your just going to play on your computer (ie. no changes to the atucal sound) then it would be your best option. Easiest way to use replaygain is foobar2000.