Over the last few years, the rules have changed surrounding what you can and cannot do when traveling with pets. The changing rules can be traced back to a peacock traversing through Newark airport.
A few years ago, the skies were filled with flying "emotional support animals." Like all good things, people abused it and ESA animals became a giant grift.
It all came to a head when a woman tried to bring an emotional support peacock onto a plane at Newark airport.
After that incident, legislation got passed and the airlines banned ESA animals.
You can travel with your dog, cat, or household bird in the main cabin, but it has to fit under the seatback in front of you. The suggested size is 11" x 18" x 11". Each airline has fees associated with traveling with a pet in cabin.
Should you want to travel with a larger animal, they will have to be a service animal. The Travel Nerds podcast covers the topic well.