Most people know of Moritz, Manfred von Richthofens beloved Great Dane (he was most likely a mutt but MvR swore up and down he was a purebred). After MvRs death Moritz got taken care of by Manfreds old friend Alfred Gerstenberg.
But there are several other pets (primarily dogs) that deserve a mention as well as many of them accompanied their owners to the front.
One of them is Max Immelmanns Great Dane Tyras. Immelmann found him in an animal shelter and his name is technically a pun, as Max named him after the place he got him from (Tierasyl = Tieras = Tyras, which was also the name of Otto von Bismarcks dog). Tyras was very loved, he was regularly sketched by Immelmann and was allowed to sleep on the bed. Tyras most likely stayed with Immelmanns family after his death.
The great Oswald Boelcke also had dogs. Besides horses they were his favourite animals. Shortly before the war broke out he got Ibi, an Airedale Terrier. He left Ibi at home during the war. Missing his loyal companion he got himself a German Shepherd named Wolf.
Rudolf Berthold got gifted a small white terrier by comrades, name unfortunately unknown. But important enough to be included in the Sanke card.
Ernst von Althaus' beloved dog Mousse was with him during the whole war. He seems to be a type of hunting breed. With him in the picture is Otto Bernert with a Jack Russel.
Josef Jacobs also brought his dog Rolf to the front with him.
Heinrich Gontermanns dog also had the honour of being included in the Sanke card.
Richthofens former flight companion Graf Erich von Holck also had a small dog, who he took with them in the airplane during their patrols over Russia. That tough doggy survived the incident where they flew over a burning town and had to do an emergency landing but then got lost on the way back. Holck apparently was also a cat lover. There are several pictures of pilots (from all countries) with lion cubs, so there seemed to be a trend.
Speaking of rather curious choices of animal companions, Ernst Udet and his comrades entertained a monkey named Betty. Unfortunately, I can't find the picture right now (does anyone have it?).
Ernst Udet tells the story of his comrade Wilhelm Klinkermann whose dog always waited right at the flight strip until his owner returned. On the day of his death the dog was very nervous and when Klinkermann didn't return, they had to force the dog into the car to take him to his family. He did not want to leave without his owner.
Walter von Bülow-Bothkamp with his dog Tönnes.
And here some more (I'm not sure if these dogs belong to whom they are pictured with):