Lufthansa carries small pets in the cabin and larger ones in the temperature-controlled hold on most long-haul aircraft, with clearly published weight and crate rules and a strong track record for animal handling.
In-cabin travel
Allowed Checked (hold) travel
Allowed Manifest cargo
Allowed IATA Live Animals Regulations: IATA LAR / CR82
Comparing carriers for a route?
Airline pet rules stack on top of the destination rules. Check both before you book.