I had a home inspection today and was told that some of my ceiling mounted lighting fixtures need electrical boxes. My house was built in 1844 and these boxes weren't used then. I was also told that I had some outlets that were reverse polarity. Any idea on how to remedy this situation. I am running 100 amp service and have the old romex two wire system.
Thanks

How did they know there are no boxes? H-I's are NOT supposed to remove fixtures, etc. At least in my experience.
In any case, if the fixtures are original to the houses being wired then it is what it is.
If someone mounted a modern fixture to an old iron box than you DO have violations.

Reverse polarity means hot and neutral are reversed.

The H-I opened up a floor access panel in one of my closets upstairs. When he opened the panel the wiring for the ceiling fixtures was right there, just kind of laying there with no type of "box" that is supposedly supposed to be connected to.

I bet there was a box, he just couldn't see it from above.

For a house that old, that hasn't been completely re-wired, I would say sell it AS-IS. Let the H-I say whatever he wants. You say it is an old house and is being sold as such. If the buyer wants to re-wire it after they move in so be it.
There are thousands of homes like that in my area. They get bought and sold all the time.