Many Korean and Japanese dramas are based off of Anime or Manga so the stories are popular in both countries.
Two of the most popular Japanese dramas also have Korean ones!
Boys Over Flowers v. Hana Yori Dango
Hardworking Geum Jan Di attends the exclusive Shinhwa High School by a scholarship she gets because of an incident. She finds her shallow, privileged classmates unbearable, especially the notorious F4. Angry at their rudeness, Jan Di tells off F4's leader Jun Pyo, bringing hell along with her. The proud and persistent Jun Pyo falls hard for feisty Jan Di, but she shares a connection with the quiet, sensitive Ji Hoo.
Here are the two different trailers :D
To The Beautiful You v. Hana Kimi
The main character is Ashiya Mizuki, a Japanese girl in the United States who one day sees the young athlete Sano Izumi compete in the high jump on television. She begins to idolize him. Sano gets injured one day in America while trying to help Mizuki escape from some gangsters. He stops high jumping ever since. Mizuki blames herself for the incident and decides to go to Japan and help Sano find his love for high jumping.
Izumi goes to an all-boys school, so Mizuki disguises herself as a boy to meet with him again.