when i start the car, it runs fine but when i tried to turn the engine off, the engine still runs so I had to take the bypass wire going to the starter, off and then put it back on. I had to take off the negative cable from the battery to prevent from losing power in the batter. What should I do? The mechanic at a foreign shop said it may need wiring harness replacement which may cost up to $600 or higher. I do not believe him so what are your recommendations and save money?

This car definetly is in the range of the dreaded failing wireing harness years. The insulation fails on the enginne harness on early to mid 80's Volvo's.
Take a look at the grey connectors on the firewall and the connectors near the alternator. Is the insulation cracked and easy to crumble in your fingers. If so yours is in need of replacement.
Gary

After checking with the wiring harness, it is in good shape. I checked the cables from the alternator. It looks good. I get a feeling that the wiring coming from the ignition coil may be malfunctioning. Would something coming from the ignition coil need replacement. My car still runs when I try to turn off the engine. I have to take the wire off the ignition coil and shut the engine off. I have been doing this for two weeks now. What to do next?

Did you check the wires at the starter? That's where mine had shorted together causing it to keep running.
If it's not obvious at this point it will take a wireing diagram and some serious troubleshooting.
Gary

Did you check the wires at the starter? That's where mine had shorted together causing it to keep running.
If it's not obvious at this point it will take a wireing diagram and some serious troubleshooting.
Gary

Yes, that is where I am suspecting...I am a little worried of the costs for that kind of repair. What is the cost of the wiring problem with the starter? THe mechanic said it could be the wiring harness or a bad starter. The recent owner of the car said it is not the wiring harness as he knows the car very well. It may be the wiring shortening part..WHat do you think?

SOmetimes when I turn off the ignition switch, it turns off normally but at times, the engine runs when the switch is off. Is it something around the ignition switch? The recent repair was that the mechanic was doing the bypass wiring that starts from the ignition coil into the instrumental panel, which I assume that leads to the ignition switch. I had to unplug the bypass wire from the ignition coil that shuts off the engine completely. Get what I am saying?

I know that the engine harness can cause the problem you describe, it hapened to me. The wires near the starter had shorted together. I can't say whether or not it is your specific problem. Check the condition of the harness. If it is OK then I would check the ignition switch. The only way I know to check the ignition switch is to remove and examine it.