It is certainly possible to alter the facial appearance of a person to make him/her resemble another person using prosthetic make-up.
But those shown in the movies such as Mission Impossible, Charlie's Angels and the one depicted above are not practically possible as it would involve the mask to be a one-piece wearable and removable item, very much unlike current date prosthetic masks that are a composition of a set of different pieces.
To gain a little more perspective of what I am talking about you could go through this slide that shows the transformation of Mr. Amitabh Bachchan for the Bollywood movie Paa.