Men dyeing hair:[1]

It is [Biblically[2]] forbidden for a man to dye white hair to a black color [or other beautifying color[3]].[4] [It is likewise forbidden to dye black or other colored hair to a different beautifying color.[5]] This prohibition applies to even a single hair.[6] [This prohibition applies even to a temporary dye.[7] This prohibition applies even in a time of need, even if not dyeing the hair will make one look shameful and embarrassing.[8] Some Poskim rule that in areas that men are accustomed to dye their hair then it is permitted for men to dye it, as explained in Q&A!]

Black to white-a degrading color:[9] It is permitted even initially for even men to dye a black hair to a degrading color, such as a white color.[10]

Women: It is permitted for a woman to dye their hair without any restriction.

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[1] Michaber 182/6; Tur 182; Rambam Avodas Kochavim; Ohel Yaakov p. 382-391]

[2] Tur 182 in name of Rambam that one who does so is liable

[3] Shevet Hakehasi 2/257-2

[4] Michaber ibid; Tur ibid; Rambam ibid

The reason: As doing so is a beautification that is done by women and is forbidden due to the prohibition of Beged Isha. [Taz 182/7; Biur Hagr”a; Beis Yosef 182] It is done in order so one look young and handsome, hence falling under the above prohibition. [Beis Yosef ibid]

[5] Chelkas Yaakov 87; Mishnas Yosef 8/17; Shevet Hakehasi 2/257

[6] Michaber ibid; Tur ibid; Rambam ibid

[7] Shevet Hakehasi 2/257-2; Poskim brought in Ohel Yaakov 79 footnote 107

[8] Darkei Teshuvah 182/17 in name of Poskim

[9] Taz 182/7; Beis Yosef 182 in correct Girsa of Rambam ibid; See Mishnas Yosef 8/17 regarding if the dye does not make the person look more handsome.

Other opinions: The Tur writes in name of the Rambam that one who dyes a black hair white is liable. The Beis Yosef and Taz ibid attribute this to a misprint, as in truth there is no prohibition involved in such a case.

[10] The reason: As doing so is not a beautification that is done by women, as it is done to make one look old and respectable. [Beis Yosef ibid]

Other opinions: The Tur writes in name of the Rambam that one who dyes a black hair white is liable. The Beis Yosef and Taz ibid attribute this to a misprint, as in truth there is no prohibition involved in such a case.