The Old Testament law forbade priests from shaving their heads or beards (Leviticus 21:5). In the ancient Hebrew culture, men usually had long beards. In fact, it was considered a disgrace for an adult man not to have a beard (2 Samuel 10:4-5). Men taking the Nazarite vow were not to cut their hair until the vow was completed, at which time they were to shave their heads. Regarding women shaving, Scripture says that it is disgraceful for a woman to shave her head (1 Corinthians 11:5-6) and that her hair “is a glory to her” (v. 15 KJV). Other than the hair and beard, the Bible nowhere directly mentions shaving any other part of the body.
It appears that God has given us great freedom in regard to what we do with our hair. The only New Covenant principle that applies today is that men should have masculine hairstyles and women should have feminine hairstyles (1 Corinthians 11:3-16). Even in that, there is a lot of flexibility, as what differentiates masculine and feminine varies somewhat from one culture to another. Hair is interesting in that, for humans, it does not serve a critical purpose. A human being could live a healthy life with no hair whatsoever. The fact that hair grows back when cut or shaven likely indicates that God intends us to have hair. The fact that hair is so easy to adjust and alter likely indicates that God allows us to shave, cut, style, etc., according to our personal preferences.
There is a trend today of people shaving their entire bodies. Men and women are shaving their heads, legs, arms, armpits, chest, eyebrows, and private areas. Laser hair removal, which is permanent, is becoming increasingly popular. While it is in the realm of personal preference and individual freedom, completely ridding our bodies of hair does not seem to be consistent with the fact that God gave us hair for a purpose. While a person is biblically free to shave whatever area of his/her body he/she wants to shave, the body-shaving trend may be more about vanity than preference or necessity. That is the only issue a person should consider in regard to shaving.