Ghosts definitely exist. I'm not sure what they are, for all I know the old Stone Tape idea is valid after all. But I and many members of my family have had multiple experiences dealing with ghosts, and there's really no logical, scientific explanation for those experiences. Multiple people seeing the same thing can't be reasoned away as the mind playing tricks or a hallucination. Seems to show up with very young children more than adults, though, especially around age 2.
What's weird is that most of my experiences weren't frightening, to such an extent that I actively thought "why aren't I more freaked out about this" while staring at my dead grandmother hovering in the hallway. There just wasn't any emotional reaction. Then the she was gone and I sort of shrugged and went back to my room.
Aside from actually seeing a full blown manifestation, maybe the weirdest experience I've had was about a decade ago. I was talking about our haunted house to a younger cousin, and specifically mentioned that objects tend to disappear (always in the Spring months for some reason) and will reappear if we tell my great-grandpa to put them back.
My cousin suddenly noticed that her cellphone was gone from the table where she put it and we couldn't find it. We spent a solid hour searching the house for it, and despite it having a very obvious bright pink case, we couldn't find it. I finally had her ask gramps to put it back, and literally turned around and it was sitting on the piano, exactly in my eyeline. Zero possibility that I didn't see it, as I was searching that room repeatedly moments before. She wasn't in the room at the time, she was out in the hall freaking out, so even if she had hidden the phone herself to mess with me, she couldn't make it reappear like that.