I'll do a write up later.. but in the mean time their MO is to get you to look at your Event Viewer for errors (there are always) errors, and then visit ammyy.com to run some remote control software. DO NOT LET THEM DO THIS!
I know this type of scam is quite common, and ammyy.com even admits that it is often abused in this way. There was a degree of sophistication here though in that they had a close approximation of my wife's name and we have an unlisted telephone number.
There were two operatives, the first one handles the initial part of the call and makes you open up your Event Viewer to look for errors and warnings (there are always some of those) and then warns you not to open the warnings or you will damage the computer. Operative number one had an Indian accent and sounded like they were coming in over a voice-over-IP connection.
After mucking the operator around for 20 minutes I confronted them with what they were doing. He was unapologetic and full of bullshit, and was still trying to connect to my machine.
Of course, the whole thing is a scam. I don't have a support contract for my version of Windows, the errors in my Event Viewer were harmless.. but if I had let the operator take control of my machine then he could have installed any sort of malware on it, or trashed the machine and then charge me a fortune to fix it.
I've been working in the IT field for almost 25 years and frankly it was obvious in the first few seconds that this was a scam. But for a naive user it might seem credible. If (like me) you end up doing tech support for your relatives, it might be a good idea to edit the PC's hosts file to block ammyy.com and www.ammyy.com: