One of the most appealing characteristics of a leopard gecko is the fact that they seem to almost recognize their owners. They are very alert animals and always very attentive to what is going on around them, and many times they will wander right over to your hand when you place it in the cage, possibly even climbing right onto it. While this may just be their way of searching for food, it is certainly endearing. If your leopard gecko is friendly toward you and seems not to mind being touched, you may want to take him out of his cage and hold him from time to time. However, if he resists being touched or runs away, leave him alone for a while and try again another day.
One of the most important things to remember when physically interacting with your leopard gecko (or any of the objects in his cage) is to wash your hands thoroughly before and after any contact.