Imam Musa al-Kazim (a) is reported to have had a number of debates and discussions with Abbasid caliphs and with the scholars of other religions and of Muslim denominations. His reported debates with Jewish and Christian scholars were in response to their questions and have led to their conversion to Islam.
Conversation with Abu Hanifa
It is reported that Abu Hanifa once came to visit Imam al-Sadiq (a) to ask him a question. He first met Imam al-Kazim (a), who was five years old at that time. Abu Hanifa asked him the question that he had in mind: Is the sin caused by God or by people? Imam al-Kazim (a) responded: "[There are four possibilities:] it is either from God without people having any role in it, in which case God would not punish them because of what they did not have a role in; or it is caused by God and people, in which case God is the more powerful partner in sin, and it is not right for the more powerful partner to punish his less powerful partner for something they committed together; or it is caused by people without God having any role in it, in which case God would forgive them if He wants or otherwise punish them." When Abu Hanifa heard this response, he decided to return without meeting Imam al-Sadiq (a) as he found the response to be adequate.
In another occasion, it is reported, Abu Hanifa told Imam al-Sadiq (a), "I saw your son Musa praying while people were passing by in front of him and he was not preventing them." Imam al-Sadiq (a) asked his son about what Abu Hanifa had said, Imam al-Kazim (a) said in response, "Yes, because the One for whom I pray was nearer to me than those who passed by" and adduced the verse "We are nearer to him than his jugular vein" (Qur'an 50:16).
Conversations with non-Muslim Scholars
With Jewish Scholars
According to a hadith, a group of Jewish scholars asked Imam al-Sadiq (a), "What is the miracle of your prophet?" The Imam (a) touched the chest of his son Musa (a), who was five years old at the time, and told the Jewish scholars to ask him their questions. The Jews asked Imam al-Kazim (a) about the signs that was given to Prophet Muhammad (s). The Imam counted many miracles and at the end mentioned the Qur'an and stated that it had miraculous characteristics that would require a long time to describe. They asked, "How can we know that it is as you describe?" Imam al-Kazim (a) responded, "How can we know that what you say about the miracles of Moses are as you describe?" They said, "We know that by referring to the reports of trustworthy people." Imam al-Kazim (a) said, "Then know the truth of what I tell you by the report of a child whom God instructed without the instruction [of anyone else] and without any knowledge acquired from the reporters." At this moment, the Jewish scholars, who were amazed by the knowledge of Imam al-Kazim (a) despite his young age, embraces Islam and testified that Ahl al-Bayt (a) were the true Imams and Proofs of God. Then Imam al-Sadiq (a) kissed the forehead of his son and said, "You are the one who rises [as an Imam] after me." 
According to a report, Burayha, a prominent Christian figure, together with a group of other Christians had a debate with Hisham b. al-Hakam, in which Hisham defeated them, so they went together with him to Medina to meet Imam al-Sadiq (a). When they arrived, they met Imam al-Kazim (a) at the door. Hisham told the Imam (a) the story of the debate and then Imam al-Kazim (a) had a conversation with Burayha and recited parts of the Gospel for Burayha in a perfect manner. This amazed Burayha and thus he embraced Islam together with his maid. Hisham, then, told the whole story to Imam al-Sadiq (a), and the Imam (a) recited the following verse: "Some of them are descendants of the others, and Allah is all-hearing, all-knowing" (Qur'an 3:34).
It is reported that on one occasion, when Imam al-Kazim (a) was fleeing the persecution of the Abbasids, he entered a Syrian village and met a monk there. The monk asked him some questions and embraced Islam when he heard the Imam's responses.
According to another report, Imam al-Kazim (a) had a conversation with a monk. Among the exchanged questions and answers were the following: The monk asked the Imam (a), "How do you say that the Tuba tree has its roots in the houses of Jesus (a) and Muhammad (s) and its branches in all houses?" The Imam (a) responded, "Like the sun, whose light enlightens all places while it is in the sky." The monk asked, "How could heavenly fruits and food neither end nor decrease?" The Imam (a) responded, "Like a flame, which produces other flames without its own light decreasing."
Conversations with Abbasid Caliphs
On another occasion, Imam al-Kazim (a) requested that al-Mahdi return Fadak, a village that the Prophet (s) had granted his daughter Fatima (a) but was confiscated by Abu Bakr. Al-Mahdi asked the Imam (a) to delineate the borders of Fadak, and the Imam delinated borders that made Fadak equal to the whole territories of the Abbadids. According to another report, this request for Fadak took place at the time of Harun al-Rashid and the Imam (a) mentioned Eden, Samarqand, Ifriqiyya, and Sayf al-Birr as its borders. Then Harun said, "Then nothing remains for us," and the Imam responded, "I knew you wouldn't return it."
On one occasion, Harun al-Rashid was reportedly near the grave of the Prophet (s) with Imam al-Kazim (a), and, in order to show his closeness to the Prophet (s), called the Prophet (s), "O cousin!" In reaction to that, the Imam (a) called the Prophet "father."
According to another report, Harun al-Rashid told Imam al-Kazim (a), "Why do you let people attribute you to the Prophet (s) when you are the sons of Ali (a) and not the Prophet?" The Imam (a) responded, "If the Prophet (s) came back to life and asked you to wed your daughter to him, would you do that?" Harun said, "Yes, and that would be my honor over Arabs and non-Arabs." Imam al-Kazim (a) then said, "But the Prophet would never ask me to wed him my daughter, because he is my grandfather." Again, Harun said, "How do you consider yourselves descendants of the Prophet (s) while he did not have a son?" Imam al-Kazim (a) recited the following verses: "And We gave him Isaac and Jacob and guided each of them. And Noah (a) We had guided before, and from his offspring, David (a) and Solomon (a), Job (a), Joseph (a), Moses (a) and Aaron (a) thus do We reward the virtuous and Zechariah (a), John (a), Jesus (a) and Ilyas (a), each of them among the righteous" (Quran 6:84-85) and said, "Although Jesus (a) did not have a father, God considered him a descendant of Abraham (a) through Mary. We are also descendants of the Prophet through our mother Fatima."
Ibn Shahrashub (d. 588/1192) also reports a conversation between Imam al-Kazim (a) and Harun in Mecca, according to which Harun asked a question from the Imam (a) and the Imam (a) answered and then the Imam (a) asked Harun a question but Harun was not able to answer.
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