If people link to you incorrectly, does it hurt you?
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In Google WM tools I'm seeing so many 404 crawl errors but they're all from other sites linking to us incorrectly, which I can't do anything about. Will this hurt us somehow as far as SEO goes?
The logical thing would be that it would hurt the site doiing the linking but it does come up in OUR WM tools, so it makes me wonder.
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I'm not sure if it can hurt you, however you can redirect their bad links to working pages with 301 redirects.
example: I link to your site with yousite.com/paeg.html (misstyped on purpose) but your page is actually located at yoursite.com/page.html
In this case create a 301 redirect to point from /paeg.html to /page.html and you will get people coming to the right place.
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You can help this!

You can use the data from WM tools to create 301 redirects on your site that will direct these incorrect queries to the correct page, preserving more of the authority than letting it fall off with a 404
There was actually a post recently (4 or 5 months ago) about doing this automatically using a server side script that actually "fixes" it for you automatically. But it is pretty advanced, and if you aren't dealing with thousands of them, might be easier to do yourself (or have your webmaster take care of) (I was not able to find or remember what it was called, maybe someone else will)
If this is just PDF documents and Files that have been picked up by scrapers, I personally don't worry about these so much, but someone may disagree with that

w00t!
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If you've got inbound link to non-existent pages then set up 301 redirects to the most relevant page. At the moment any link-juice you could be receiving form these links is being wasted.
Broken links won't hurt you - you just won't get the benefits!
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Thanks for all the responses. Of course, I thought of the redirect route, but there are hundreds of wrong links, so I was trying to avoid the 301 thing. I am the webmaster, so if anyone does remember what that script was, that's be great!
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No they won't hurt you, but setup redirects for them.
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LOL,
I knew I saw it...
http://www.seomoz.org/blog/set-it-and-forget-it-seo-chasing-the-elusive-passive-seo-dream
A script engine created by virante based off of Levenshtein Distance, also known as the Edit Distance
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Thanks so much!
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Generally speaking, inbound links can't hurt you. Websites link to each other all the time and websites move or remove content all the time. 404s hurt if your own site if you are linking to bad URLs because you're wasting crawl budget.
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I think everyone hit this on the head but I had one quick point to drop. It might be good to search over the sites as well to look at the most valued sites that are linking to you with 404 errors. I would try to then reach out to them personally and ask if they could update the link. You often can get a specific anchor text or a link to a page deeper within your website.
Good luck with it!