There are pet cafes/cat cafes in Korea with adorable dogs and cats. You can touch the animals that are freely wandering in the cafe or interact with them by feeding them snacks.

Marie N Zoo
Marie N Zoo is a unique animal cafe in Songdo. There are other pets than dogs and cats, such as rabbits, guinea pigs, hedgehogs, parrots, skunks, foxes, and meerkats. Each animal roams freely in a separate room. You can interact with the animals by feeding them snacks as per the zookeeper's guidance. However, since foxes are sensitive animals, you can only enter the fox room every hour with a zookeeper. The advantage of this place is that you can meet animals that are difficult to see in the city.
* Admission fee is 10,000 won inclusive of drinks.
Address: 13, Sinsong-ro 125beon-gil, Yeonsu-gu, Incheon
Phone: +82-32-831-5437
Hours of Operation: Mon - Fri 12:00 ~ 21:00
Sat - Sun 11:00 ~ 21:00
Holidays : Open year round
Website: marienzoo.modoo.at/

Ujunyang Cat Cafe
Ujunyang Cat Cafe in Nonhyeon-dong is a cafe decorated with vertical space wherein cats can freely stay and a cat wheel necessary for cats. You can have a good time with cats of different species such as Munchkin, Scottish Fold, and Selkirk Rex. Cats here follow people, have a lot of charm, and love snacks. When you take out a snack, cats flock around you. Snacks are sold separately at the cafe.
* Entrance fee is 10,000 won inclusive of drinks.
Address: 17 Cheongneung-daero 595beon-gil, Namdong-gu, Incheon
Phone: +82-10-8513-8168
Hours of Operation: Daily 12:00 ~ 20:00

RoastingCat Cafe
RoastingCat in Bupyeong is a cat cafe where you can play with cats. Overall, it is decorated with wooden furniture, giving a warm and cozy ambiance. You can sit at a sunken table and interact with cats. You can use the toys scattered around and play with cats. If the cats are gentle, they may approach you while you are drinking.
* The admission fee is 8,000 won inclusive of drinks.
Address: 1360, Gyeongwon-daero, Bupyeong-gu, Incheon
Phone: 0507-1380-1267 (DOMESTIC CALLS)
Hours of Operation: Mon, Wen-Sun 13:00 ~ 22:00
Holidays : Closed on the second and fourth Tuesday.
Instagram: http://www.instagram.com/to1004soo/