Microsoft and other respected companies use this service to assist customers with technical issues.
However, as the service is reliable it is often used by scammers posing as Microsoft Engineers or similar. These scammers have phoned you, emailed you or you've seen a popup telling you that there is a problem on your computer - usually that it's infected. Never give them access to your computer!
More info on these scams can be found at Microsoft
These messages are NOT from LogMein, and LogMein are not involved in these scams, they just provide a service.
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Please add any others you know. Thanks to c۞g for the list above.