The two principal Japanese schools disagree about method. Rinzai, from the Chinese Linji line and established in Japan by Eisai, who died in 1215, works with the koan: a case drawn from the recorded encounters of the old masters, put to a student in private interview and answered from somewhere other than the reasoning that first meets it. Soto, brought by Dogen, who returned from China in 1227 and founded Eihei-ji in 1243, works with shikantaza, just sitting. Dogen's position is that practice and realisation are not two things.
The line between them is less clean than the summary suggests. Soto has used koans, Rinzai teaches sitting, and the third Japanese school, Obaku, arrived from China in the seventeenth century with Pure Land recitation attached to its Zen.
What the tradition has argued about hardest is the transmission itself. Manzan Dohaku, who died in 1714, succeeded in changing Soto's rule so that transmission passed from a particular master to a particular disciple rather than being certified by awakening, a redefinition of what the document attests. Hakuin, who died in 1769, and whose system organises all modern Rinzai koan practice, never received a formal transmission himself.
The twentieth century made the question urgent in the West. Richard Baker resigned as abbot of the San Francisco Zen Center in 1984 after sexual relationships with students became public. Eido Shimano, whose conduct Robert Aitken had documented and put to his society's board in 1995, retired as abbot in 2010, was barred from teaching under that society's auspices in 2011, and was publicly disowned by Myoshin-ji in 2012. The community around Joshu Sasaki admitted publicly in 2013 that it had struggled with his sexual misconduct, which reached back to at least the 1970s.
Each of those teachers held authentic transmission. That is the point the tradition has had to absorb.