I've collected some info on the topic in my website.
faster-than-light
http://www.projectrho.com/rocket/rocket3v.html
slower-than-light
http://www.projectrho.com/rocket/rocket3aj.html
The basic problem with faster than light travel is that it violates causality. Which means it more or less destroys the entire foundation of science.
The old saying is
"Causality, Relativity, FTL travel: chose any two." You cannot have all three.
Physicists hold on to Einstein's relativity quite strongly because it is been tested to more than nineteen decimal places. People have been trying to disprove it for about one hundred and five years, and utterly failed.
Physicists hold on to Causality even more strongly, because unless causes precede effects, the foundation of science crumbles.
So physicists toss out FTL travel.
FTL travel violates causality because an FTL starship and a time machine are two terms for the same thing.
The only way out is if there is some magic law of science that somehow prevents one from using a time machine to make a paradox. Read the link for details.
Years ago I got tired with all the email I got, which were all along the lines of "but maybe a scientific breakthrough will allow it, they thought man would never fly..." Now I just direct them to another section of my website.
http://www.projectrho.com/rocket/rocket3al.html