Delta is one airline that does allow small cats and dogs and sometimes small birds to travel in the cabin of the aircraft with the owner. The pet must be secured in an approved under the seat carrier. For larger pets, the pets must be check baggage or cargo and must be transported in an airline approved dog carrier as well.
To determine Delta airline approved dog carriers you can go directly to their “Pet Rules” site. The under the seat carrier is designed much like a duffle bag only it meets the requirements to safely and securely transport the hound. The bags are comfortable and well ventilated and are carried as by two hand straps or shoulder straps. Other approved styles are pet carriers that are designed with wheels and an extended handle which owners can conveniently push their animals.
Other designs include the pet crate which is designed to fit under the seat in the cabin area and for larger dogs, can be purchased for travel for the cargo area. These crates are constructed with a durable sturdy plastic and are well ventilated and include an easy to carry handle for the owner to transport the animal inn the carrier.