1000+ bad sectors is definitely cause for concern. When you said "a few" bad sectors I was thinking 5-10.
Bad sectors are a sign that the surface of the disk is becoming unreadable. I've seen quite a few hard drives develop 1-8 bad sectors within a year or two of operation. This is a bad sign, but it's primarily because a few bad sectors can lead to hundreds down the line. I've had drives with 1 or 2 or 3 bad sectors stay pegged at that number over 2-3 years of operation, with no further problems.
However, if you've already got 100 bad sectors, let alone 1000, you've got a big and probably growing problem on your hands. It means your data is at risk of disappearing.
I would follow most people's advice -- plug a second hard drive into your computer, and manually back up your high-priority files first (important documents), then your less high priority files (photo albums, other documents, etc). Keep going down the line and copy your lower priority stuff (music, movies, etc).
Now, if you want to, you can image your drive, but only once you've gotten what you want off the drive manually. I've seen people try to image a whole failing hard drive and inevitably the image fails midway through because sectors can't be read, and sometimes the failing drive then becomes unreadable.
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Finally, I'd like to add one more thing. After you've gotten what you need off the drive, it's still a good idea to format it and either write the drive with junk files or use a "file shredder" program to erase your files securely (by writing the drive with 0's or randomly generated junk).
While your drive may become unstable to read, it could still be read by someone else down the line if you don't personally destroy the disk. Most people have files saved on their computer that they don't want other people to access. It's a good idea to clean your disk by writing it with junk or a file shredder program so someone in the future doesn't have access to all of your documents/files/etc.
this is indeed superior advice.
1. Back up most important files, manually.
2. Back up the rest of the files you want to keep, manually.
3. Throw current drive in the trash (or get it replaced under warranty) and get a new one.
DO NOT UNDER ANY CIRCUMSTANCES IMAGE THE DRIVE!
1. Imaging attempt greatly increases the chance of the drive failing without you getting your important data back.
2. If you had already backed up your important data, imaging the drive will give you an OS install with CORRUPT OS FILES. This will cause bizzare and unique and annoying things to happen. You want to do a fresh install from the windows / linux DVDs not a clone with corrupt files. And you want it on a new drive.