What Finger Are They Exspose To Go On After The Wedding?

When I got married, i had both rings on my right hand. I always had my engagement ring than my wedding band. I was always told that the ingagement ring was from his heart and the wedding ring repersents the promise. I was told different stories about it. They said " place wedding band before engagement ring., or place the wedding band after the engagement. Etc...... Does it really matter what way it goes?