Both men and women shake hands. If I meet a colleague or professional associate for the first time - whether it's a man or a woman - the normal convention is to shake hands. In a business setting, everyone shakes hands with everyone, at least the first time you meet. In social settings, what tends to happen is that men shake hands with men, while women share a peck on the cheek or brief hug with both other women and with men.

In the U.S. in a professional situation you shake hands, regardless of gender. If two groups meet--e.g. if a group of five people from company X come to company Y for a conference with five people from company Y--there may be just a few handshakes--perhaps just the spokesperson from each group will shake, or perhaps each person from company X with will shakes hand with the closest person from company Y, rather than lining up so that every person from X actually shakes hands with every person from Y.

It was said in a Russian etiquette book I read some years ago that men must wait if a woman offers her hand or not. And a woman may chose what is convenient for her. But men should never stretch out their hands first.

Yes, in the U.S., professionals shake hands when they first meet each other, men and women alike. If you have an on-going relationship, like with a client where you are on-site and you see him/her every day, you would not shake hands every day. If, on the other hand, you visit the client once a month or less often, you would probably offer your hand when you meet again and exchange ""hello, how are you". These days, you don't shake hands from a seated position, even for a woman. If seated, you would rise to greet someone that you were meeting. Women now give (and expect) a full-hand handshake, not just the fingers.

In professional settings, everyone shakes hands - men and men, women and women, men and women, etc. In social settings, you can shake hands with a quick "nice to meet you." It depends how casual the social setting is. You can hug, air kiss on the cheek, or just wave. But in a social setting, you don't have to greet everyone by touching them, haha.