In the movie "Shaft," which starred Samuel L. Jackson as the title character, Shaft is looking for a certain waitress who is the only witness to a murder. The only problem is, he can't find said waitress. She seems to have disappeared from society or dropped off the grid.
Dropping of the grid is always risky because you are essentially giving up your life as you know it. You cannot go back to your old way of life lest you want to be found by whomever it is chasing you. You have to sever you connections with society (family, friends, and co-workers).
How does one just disappear off the face of the earth? Is it as simple as moving away to another country? First you have to understand who you are running from. If you are running from an abusive partner, know that they have limited resources and might have a problem if you move out of state or out of the country. If you are running from the FBI for example, understand that these people have unlimited resources and will do everything in your power to hunt you down even if you move to another country.
Before going underground, you have to do a few things in order to throw your opposition away from your trail. You have to destroy the "old" you and build a new you. Destroy all photographs you can get possession of. Some of your friends or relatives might be hesitant to give you their photographs but let them understand that the people chasing you might go after them, too, because of their connection to you. They will threaten your loved ones with physical or other forms of abuse if they refuse to hand over photographs. If it is not possible; destroy all the most recent pictures of yourself and leave the old ones. Of course you don't want your face plastered all over the state but if it is an artist's rendition or just an old photo, chances are high that people won't recognize you.
Destroy or throw away all your credit cards and ATM cards. Government authorities can pinpoint your location within seconds if you use your magnetic cards. Before going underground, be sure to withdraw all your money from the bank and max out all your credit cards with the things you'll need or you can resell for cash. Do not bring your car with you. That will be the first thing that the opposition will look for. Leave your car in place where car thieves will be and let them strip your car clean to remove any trace of it. If you need transportation, there are a lot of places where you can simply give money and get a new car without any identification required. Just make sure you aren't buying a stolen car or the car itself doesn't violate any safety rules. The last thing you want as you are making your escape is to be pulled over for a simple safety violation.