Grandfather paradox
The grandfather paradox is a very simple, science-fiction-based apparent inconsistency at the very heart of the idea of time travel into the past. It's very simply that you travel into the past and murder your own grandfather before he sires your mother or your father, and where does that then leave you? Do you instantly pop out of existence because you were never made? Or are you in a new causality scheme in which, since you are there you are there, and the events in the future leading to your adult life are now very different? The heart of the paradox is the apparent existence of you, the murderer of your own grandfather, when the very act of you murdering your own grandfather eliminates the possibility of you ever coming into existence.
Among the claimed solutions are that you can't murder your grandfather. You shoot him, but at the critical moment he bends over to tie his shoelace, or the gun jams, or somehow nature contrives to prevent the act that interrupts the causality scheme leading to your own existence.