He studied with his father, and also in yeshivas located in Slutsk and Shklov. He also had a close relationship with his uncle, Yaakov Kantrowitz, rabbi of Timkovichi, whom he greatly revered and considered his mentor.
He was appointed rabbi of Lubań, where he served for sixteen years. He married Shima Kustanovich in 1920, and had 4 children (Pesach Chaim, Fay Gittel, Shifra, and David), before leaving Europe. His son, Pesach Chaim, died in Europe, and his son, Reuven, was born in the US. Under increasing pressure from the Soviet regime, he moved with his family to New York City in 1936, where he lived for the rest of his life.
Owing to his prominence as an adjudicator of Jewish law, Feinstein was asked the most difficult questions, in which he issued a number of innovative and controversial decisions. Soon after arriving in the United States, he established a reputation for handling business and labor disputes. For instance, he wrote about strikes, seniority, and fair competition. Later, he served as the chief Halakhic authority for the Association of Orthodox Jewish Scientists, indicative of his expertise in Jewish medical ethics. In the medical arena, he opposed the early, unsuccessful heart transplants, although it is orally reported that in his later years, he allowed a person to receive a heart transplant (after the medical technique of preventing rejection was improved). On such matters, he often consulted with various scientific experts, including his son-in-law Moshe David Tendler, who is a professor of biology and serves as a rosh yeshiva at Yeshiva University.
As one of the prominent leaders of American Orthodoxy, Feinstein issued opinions that clearly distanced his community from Conservative and Reform Judaism. He faced intense opposition from Hasidic Orthodoxy on several controversial decisions, such as rulings on artificial insemination and mechitza. In the case of his position not to prohibit cigarette smoking, though he recommended against it and prohibited second-hand smoke, other Orthodox rabbinic authorities disagreed. Even his detractors, while disagreeing with specific rulings, still considered him to be a leading decisor of Jewish law. The first volume of his Igrot Moshe, a voluminous collection of his halachic decisions, was published in 1959. He made noteworthy decisions on the following topics:
Brain death as an indication of death under Jewish law (YD II:146,174, III:132, IV:54)
Cholov Yisroel Permitted reliance on U.S. government agency supervision in ensuring that milk was reliably kosher, and it is as if Jews had personally witnessed it (YD I:47). This was a highly controversial ruling disputed by prominent peers of Feinstein.
Separation of conjoined twins who were fused all the way from the shoulder to the pelvis and shared one heart. It is during this case that C. Everett Koop, the 13th Surgeon General of the United States, said "The ethics and morals involved in this decision are too complex for me. I believe they are too complex for you as well. Therefore I referred it to an old rabbi on the Lower East Side of New York. He is a great scholar, a saintly individual. He knows how to answer such questions. When he tells me, I too will know."
Note: Responsa in Igrot Moshe are cited in parentheses
Moshe Feinstein's grave
Feinstein died on March 23, 1986 (13th of Adar II, 5746). It has been pointed out that the 5746th verse in the Torah reads, "And it came to pass after Moshe had finished writing down the words of this Torah in a book to the very end." (Deuteronomy 31:24). This is taken by some as a fitting epitaph for him.
Feinstein invested much time molding select students to become leaders in Rabbinics and Halacha. Most are considered authorities in many areas of practical Halacha and Rabbinic and Talmudic academics. Some of those students are:
Nisson Alpert (1927–1986), Rav of Agudath Israel of Long Island, New York
Feinstein's greatest renown came from a lifetime of responding to halachic queries posed by Jews in America and worldwide. He authored approximately 2,000 responsa on a wide range of issues affecting Jewish practice in the modern era. Some responsa can also be found in his Talmudic commentary (Dibrot Moshe), some circulate informally, and 1,883 responsa were published in Igrot Moshe. Among Feinstein's works:
Igrot Moshe; (Epistles of Moshe); pronounced Igros Moshe by Yiddish speakers such as Feinstein himself; a classic work of Halachic responsa. Consisting of 7 volumes published during his lifetime and considered necessary for every rabbi to have. Of these, the final, seventh volume was published in two different forms, the resulting variations found in a total of 65 responsa. An additional 2 volumes were published posthumously from manuscripts and oral dictations that were transcribed by others.
Dibrot Moshe (Moshe's Words); pronounced Dibros Moshe by Yiddish speakers such as Feinstein himself; a 14 volume work of Talmudic novellae with additional volumes being published by the Feinstein Foundation and being coordinated by his grandson, Mordecai Tendler.
Darash Moshe (Moshe Expounds, a reference to Leviticus 10:16), a posthumously published volume of novellae on the weekly synagogue Torah reading. [Artscroll subsequently translated this as a two-volume English work.]
Kol Ram (High Voice); 3 volumes, printed in his lifetime by Avraham Fishelis, the director of his yeshiva
Some of Feinstein's early works, including a commentary on the Talmud Yerushalmi, were lost in Communist Russia, though his first writings are being prepared for publication by the Feinstein Foundation.
^Rav Yaakov Breisch in Chelkas Yaakov Vol.2 ch.37 stated that "all of his rationales are not sufficient to contradict a clear ruling of the Shulchan Aruch and halachic authorities...." Later in ch.37 and 38, Breisch extensively disputes various premises underlying the rationale for Feinstein's lenient ruling. See also Shu"t Beer Moshe Vol.4, ch.52, Kinyan Torah 1:38 for a more detailed listing of the many authorities disputing Feinstein's reasoning and conclusion.
^פיינשטיין, משה. אגרות משה חלק ז - חושן משפט חלק שני סימן ל. p. 244. Retrieved November 12, 2017. הנה בדבר שאלתו על כותב האג״מ מה ששמע שבישיבות מתירין להתלמידים לגנוב את התשובות להשאלות במבחני הסיום שעושה המדינה (רידזענם) 'כדי להונות ולקבל את התעודות שגמרו בטוב, הנה דבר זה איסור לא רק מדינא דמלכותא אלא מדין התורה, ואין זה רק גניבת דעת שג״כ אסור כדאמר שמואל בחולין דף צ״ד ע״א שאסור לגנוב דעת הבריות ואפילו דעתו של עכו״ם וכ״ש הכא שהוא גניבת דעת לכולי עלמא אף לישראל, אלא 'דהוא גם גניבת דבר ממש דהא כשירצה לפרנסתו במשך הזמן להשכיר עצמו
^See Negiah, section entitled "Shaking Hands in Halacha," for a discussion regarding Rav Moshe's opinion on this topic, both with regard to initiating a handshake and with regard to returning a handshake (i. e., where the other party extends his/her hand first). For a translation of R' Moshe's three Teshuvos (responsa) on men shaking hands with women, see 
^E. g., see Sinclair, Daniel. Jewish Biomedical Law 2004
^See RCA decision and, earlier, RCA Roundtable. (Statement by Saul Berman, Reuven Bulka, Daniel Landes and Jeffrey Woolf.) "Proposal on smoking" (unpublished) July 1991.
__________. Rabbi Moshe Feinstein on the treatment of the terminally ill." Judaism. Spring 37(2):188–98. 1988
Rabbi Mordecai Tendler, interview with grandson of Rabbi Feinstein and shamash for 18 years.
Warshofsky, Mark E. "Responsa and the Art of Writing: Three Examples from the Teshuvot of Rabbi Moshe Feinstein," in An American Rabbinate: A Festschrift for Walter Jacob Pittsburgh, Rodef Shalom Press, 2001 (Download in PDF format)