Quotes from the Lubavitcher Rebbe

The time has come to prepare the world for Moshiach. This includes making it a "settled place," through spreading the Seven Mitzvos.--Purim, 5747,
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[I]nfluencing non-Jews to keep their Mitzvos, the Seven Noachide Laws... will assist our task of making the world into a dwelling place for G-d, and help bring about the arrival of Moshiach.--Purim, 5747,
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"He did not create it (the world to be) a waste land: He formed it to be inhabited (and civilized)" (Yeshayahu 45:18).

This can only happen, when all humanity lives justly and righteously.

This of course is the theme of the Seven Noachide Commandments, with their accompanying laws and ramifications.

Consequently, it is obvious and self-evident, that in modern times, we must carry out the Divine Command we received through Moshe: "To compel all human beings, to accept the commandments, enjoined upon the descendants of Noach."--Shabbos Parshas Tsav, 5747,
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In order that the entire world should be orderly, it is essential, that each and every one of the "seventy nations," should be influenced, so that they will work on settling the world. It is insufficient for only one nation, to be acting properly.--Hisvaduyos 5750
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In these critical times, when nations are challenging one another, and violence is increasing in an unbelievable manner; the Jews have the power, to bring about peace in the entire world....

Ideally, a Jew should stand proudly before the gentiles, and explain to them the Seven Noachide Laws. Emphasizing that they should be carried out, not because they appear to be logically sound, but because G-d commanded them....

When a Jew carries out Mitzvos with pride, a non-Jew stands in awe of him and, hence, will not consider war.
Yud Tes Kislev, 5743,
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There is, of course, no need to emphasize to you, the importance of promoting these Seven Noachide Commandments among gentiles.

In our day and age, it does not require much imagination to realize that, by way of example, had these Divine Commandments been observed and adhered to, by all "Children of Noach," namely the nations of the world, individually and collectively; there would not have been any possibility, in the natural order of things, for such a thing as a Holocaust.

"Letter to Chaplain Brig. Gen. Israel Drazin," Erev Shabbos Kodesh Bereshis, 5747, Hal Boi HaOlam, p.395.
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To the contrary, such activity will increase the respect that the nations show us, for they will realize, that Jews care not only about their own welfare, but also about the good of all humanity, and the whole world.
Shabbos Parshas Tsav, 5747
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the Rambam rules, that this was commanded by G-d through Moshe--to all Jews, to spread these Mitzvas.
Can anyone say (contradict) this command?!
Shabbos Parshas Tsav, 5747.
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"All the gentile nations will flock to [the Beis Hamikdash], and many peoples will go...."

And similarly, also during the end of golus, "Beis Rabbeinu Shebe'Bavel" [770], "will be called [a house of prayer] for all peoples."
--Kuntres Beis Rabeinu Sheb'BaveI," Sefer HaSichos 5752 2:475.
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Through the shluchim [in addition to their work with Jews], it is also essential, to spread goodness, righteousness, justice, and uprightness, among the nations of the world, through the fulfillment of the Seven Noachide Laws.
Hisvadayos 5752
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The Seven Laws must be explained in a way, that the nations can relate to; and because, non-Jews will be willing to listen more quickly and easily, than a Jew.
Hisvaduyos 5748
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When the gentiles see how the Jews, despite all difficulties, follow the path which they were commanded by G-d, they will grant greater assistance to the Jews, in order to facilitate the completion of their mission.
"Address to N'Shei Ubnos Chabad," 26th day of Elul, 5746,
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Non-Jews are also enjoined in the Mitzvah of Torah study (in the areas in which they are obligated).
Hisvaduyos 5748
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The effect on the gentile nations, is especially seen through translations of Torah--including the inner, mystical aspects of Torah, which are revealed in Chasidic thought--in the "seventy" languages of the non-Jews.

Such translations were also carried out specifically for non-Jews, so that they can study the concepts of belief in G-d, Divine Providence, etc.
Hisvaduyos 5750 1:148; Hisvaduyos 5748 4:267, fn. 54 cited in Kol Boi Ha'olam, p. 374.
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Jews have been given a special mission, to teach the gentiles the Seven Noachide Laws, which are intended to spread peace and equilibrium throughout the world.

Furthermore, a Jew is always conscious of the existence of G-d, Who controls every aspect of the world, and watches each deed performed by people.

Therefore, before one does anything to benefit from the world, he first recites a blessing, that can be heard by all those around him.

When a gentile hears a Jew make a blessing, this makes an impression on him as well. For example, it will prevent him from stealing, for he will always remember, that there is a G-d watching every aspect of his behavior.

In this manner, our acts can add to the proliferation of peace and justice in the world.--"Address to N'shei Ubnos Chabad," 26th day of Elul, 5746,
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Non-Jews should be encouraged, to pray properly and regularly.
--Paraphrase from 11th Nissan, 5745, on video.
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The light of the Menorah illuminates the public domain, and neutralizes the "rebellious footsteps."

It also provides an opportunity, to urge humanity to fulfill the Seven Noachide Laws, because they are G-d's commandments for mankind.

Life according to the Seven Noachide Principles, will be free of any rebelliousness against social laws, congruous with the message of the Menorah.
"Compilation of talks, from week of Chanukah," 5747,
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Indeed, the complete expression of G-d's Kingship, is when G-d is accepted as King; not only over the Jewish people, but over the entire world.
1st Night of Sukkos, 5757,
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The end purpose of Creation is that (not only we Jews, but also) the whole world, all mankind, all children of Adam, the first man, indeed, the whole of Creation, nature itself; should acknowledge, that the Creator of the world, is the "King of all the earth."
"Letter to B'nai Yisrael," Chai Elul, 5746.
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This is associated with the Haftorah read this morning which mentions the Messianic prophecy, describing how the holiday of Sukkos, will be celebrated by all the nations.

Similarly, our celebration of Simchas Bais Hashoevah, will also have an effect on the gentiles. Not only will they not disturb the celebrations, they will assist in preparing the celebrations.

Furthermore, they will be inspired to participate, by standing on the side and clapping and the like.
2nd night of Sukkos, 5747
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This also implies involving others (even gentiles), in the celebration.

Not only must they witness the rejoicing, but they must also take part in the celebrations, and attain the unique level of jo,y appropriate for the present evening.
6th night of Sukkos, 5747.