Yagyu-ryu refers to the family sword style of the Yagyu family. Although in general usage people refer say Yagyu-ryu, it is actually not one style but an umbrella word for a collection of martial traditions that can all trace their lineage or techniques back to the Yagyu family line, regardless of whether there is anyone practicing the style with the Yagyu family name.
The generally accepted head of the Yagyu clan is said to be Yagyu Muneyoshi (1527 - 1606 A.D.), also known as Sekishusai. He learned his technique from Kami-Izumi-Ise-No-Kami-Nobutsuna, who was the founder of Shinkage-ryu (New Shadow Style). When Muneyoshi incorporated Shinkage-ryu into his existing body of martial knowledge, his style became known as Yagyu Shinkage-ryu.
Yagyu Muneyoshi Sekishusai had five sons. Of the five, only the youngest, Yagyu Munenori (1571 - 1646 A.D.), is worthy of note. He mastered the techniques of his father, then added to these with his own battle experiences, as well his deep enthusiasm for zen Buddhism. The result was a new style retaining some of the essence of his fathers art, but containing also a deeper purpose in the form of the search for self-enlightenment.
Yagyu Munenori went to Edo (Tokyo) to instruct the Shogun Tokugawa Ieyasu in the art of swordsmanship. Because of this, Munenori's style is generally called, Edo Yagyu Shinkage-ryu.
Yagyu Munenori's eldest brother, a samurai of unknown abilities, was
crippled in battle and could no longer carry on the martial tradition.
His son, Toshiyoshi Hyogo-no-suke, learned swordsmanship from his
grandfather Yagyu Muneyoshi Sekishusai. He learned the teachings
verbatim and was able to retain the katas in their original form, as his
grandfather had taught. Yagyu Toshiyoshi's professional obligations led
him to settle in a section of central Japan called Owari, near what is
modern-day Nagoya. He continued to practice swordsmanship, and handed
down what is now known as Owari Yagyu Shinkage-ryu.