First, download all of your website's files, via FTP or SFTP or the method recommended by your web host for moving files back and forth. If you don't know, then you probably had a designer do all of this work for you. Get your designer involved in moving your site. If you are no longer in touch with your designer, you'll need to dig up the username and password for your web hosting account and learn enough basics to use them. If you don't have that account information, you will have to contact your web host and prove your identity to them in another way. See my article I took over a website. How do I change it?
Once you have signed up for hosting with a new provider, just upload the files you downloaded from the old provider. If you have a database, you will also need to read your backed-up database data into the database on the new server. Again, see the article Using PHP To Back Up MySQL Databases for information on how to restore a MySQL backup.
When Your Old Host Does Not CooperateHere I've assumed that your old web host is cooperating, at least until they know what's up. But what if you don't have the login information for your old site, and your host won't give it to you?
In this case, consider mirroring your site. "Mirroring" is saving a copy of your site with a program that automatically "crawls" through the site, much like a search engine indexing program does. For more information on how to mirror a website, see my article how do I mirror another website?
Once you have mirrored the site, you can upload those files to the new provider. You will probably have to fix links and deal with some breakage as websites are not necessarily designed to be convenient to mirror in this way.
If Your Website Is Already GoneIf you have cancelled service with your old host already, they have probably deleted your files. That's the professional thing to do, after all— those files don't belong to them.
However, if you're desperate, you can try asking them to reinstate (and pay for) your account. Then you may discover that they are able to locate your files after all. Just promise me you'll keep a backup of your website's files next time!
Moving Your Domain NameWhat if your old host registered your domain for you (never a good idea, by the way) and you don't have the login information to change your DNS server to the new host?
Hopefully, when they registered your domain name, they did so in your name and listed you as the registrant in the whois database. If that is the case, then the easiest solution is to visit any domain name registration company other than the one you were previously registered with, and transfer your domain registration there. Since they want new customers, they will be happy to do that for you.
If your domain is not registered in your name— if it is registered in the name of your old host— you may, unfortunately, have to take legal steps to resolve the situation. I am not a lawyer and cannot give legal advice.
You may find that a brief letter from a real lawyer is sufficient motivation to convince your old host that pretending to own your domain name can only lead to expensive trouble.
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