There is considerable debate among the Halachic authorities as to whether or not wearing a kippah is required by law.
Jewish law dictates that a man is required to cover his head during prayer. Originally, wearing a head covering outside of the synagogue for Orthodox males was a custom.
Today, according to some Halachic authorities it has since taken on “the force of law” because it is an act of Kiddush Hashem [glorifying God]. The 17th-century authority David HaLevi Segal suggested that the reason was to distinguish Jews from their non-Jewish counterparts, especially while at prayer.
Others, however, including the Taz (commentary to the Shulchan Aruch by Rabbi David Ben Shemuel Halevi, Poland, 1586-1667), held that nowadays wearing a Kippa is required according to the strict Halacha.
Other halachic authorities like Ovadia Yosef rule that it should be worn to show affiliation with the observant community.
The Talmud states, “Cover your head in order that the fear of heaven may be upon you.” Rabbi Hunah ben Joshua never walked 4 cubits (2 meters) with his head uncovered. He explained: “Because the Divine Presence is always over my head.”
According to the Shulchan Arukh, Jewish men are strongly recommended to cover their heads, and doing so, should not walk more than four cubits bareheaded.
The Vilna Gaon says one can make a berakhah (blessing) without a kippah, since wearing a kippah is only midos chassidus (“exemplary attribute”). Recently, there has been an effort to suppress earlier sources that practiced this leniency, including erasing lenient responsa from newly published books.
According to Rabbi Isaac Klein, a Conservative Jew ought to cover his head when in the synagogue, at prayer or sacred study, when engaging in a ritual act, and when eating. In the mid-19th century, reformers led by Isaac Wise completely rejected the kippot after an altercation in which Rabbi Wise’s kippah was knocked off his head.
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