Here are some examples of the Rebbe's statements referring to himself as Moshiach:
1) The Rebbe said that as a final preparation for Moshiach, we should learn the Torah topics concerning King Moshiach and Redemption. These are found in many places in the written and oral Torah. He further said:
"One should especially learn these topics from the teachings of Chassidus, and especially in his Torah (the Chassidic discourses and Likkutei Sichos) of the leader of the generation—as a foretaste, example and preparation for the learning of the Torah of Moshiach."
The Rebbe is the only one who has written books that carry the name Likkutei Sichos. The Rebbe refers to these volumes as those of the leader of the generation, which proves that he identifies himself as the leader of our generation. The Rebbe says that "the leader of our generation is our Righteous Moshiach."
2) The Rebbe said:
"'... All the designated dates for the Redemption have passed, as the head of our generation informed us. Now is the time of, 'Please send through the hand (of whom) you will (eventually) send.' Moshe said this when he was sent by G-d to redeem the Jews from Egyptian slavery. He pleaded that G-d should send immediately the final redeemer, Moshiach, 'who will usher in the final Messianic era of peace and good.' The first redeemer who is (also) the last redeemer, the redeemer of our generation, my father-in-law [the Previous Rebbe], leader of our generation... additionally and mainly, since the aspect of 'he was victorious' has already transpired, there must immediately be the true and complete Redemption through our righteous Moshiach, leader of our generation, the Previous Rebbe; the same applies to his successor [the Rebbe is referring to himself]." The Rebbe, King Moshiach, accepted upon himself the leadership of the Previous Rebbe and became his successor on the 10th day of shevat, 5711 (1951).
3) On numerous occasions, when referring to the Redemption, the Rebbe said "that the true and complete Redemption through our righteous Moshiach will come in actuality, immediately." The word that the Rebbe uses for "immediately" is "MiYaD" in Hebrew. The Rebbe explained the significance of this word. It is an acronym for the names of the last three Chabad leaders. The Rebbe says that he starts in the order of the Rebbe closest to us:
"M - Moshiach, Menachem is his name [the Rebbe's name].
Y - Yosef Yitzchok, the Previous Rebbe, the 6th Chabad Rebbe.
D - DovBer, the second name of the 5th Chabad Rebbe."
4) There are seven Chabad Rebbes, of whom the Rebbe King Moshiach, (Menachem Mendel Schneerson,) is the seventh. Each Chabad leader was the leader not only of Chabad Chassidim, but of his entire generation. On the 10th day of Shevat, 1951, the day the Rebbe officially accepted the leadership, he said his first Chassidic discourse. It is known that the first discourse of a Rebbe contains the message of his mission and the purpose of his generation.
In this discourse, Basi Legani, the Rebbe informs us that Moses was the seventh generation from Abraham. In that generation, the world experienced an unprecedented revelation of G-dliness at Mount Sinai, when the Jewish nation received the Torah. So, too, our generation is the seventh of Chabad Rebbes, and as such, this generation brings the revelation of G-dliness to its ultimate fruition, a perfect world with the existence of only good. By the mere fact that we are the seventh generation, we merit to draw into the world the essence of the Divine presence on a higher level from that which existed at creation before the sin of the Tree of Knowledge.
From the above, it is clear that our generation merits the Redemption—the final revelation of G-dliness in the world—with the Rebbe, seventh leader, at its head.
5) The Rebbe has said many times that we are the last generation of exile and the first generation of Redemption.
6) The Rebbe meets the criteria of Moshiach and has made us aware of his identity as Moshiach. Therefore many projects have been organized to publicize the Rebbe as King Moshiach. These projects received the Rebbe's blessing and approval, indicating his acceptance of the role.
7) For over a year the Chassidim (disciples) sang "Long Live Our Master, Our Teacher, Our Rebbe, King Moshiach Forever and Ever" on a daily basis in the Rebbe's presence. The Rebbe vigorously showed his approval and encouraged those who sang.
Since it was established that the Rebbe is Moshiach, and he has told us that the Redemption is already here in actuality, we are past the stage of identifying him as the presumed Moshiach. Once he has entered the stage of actual Moshiach, there can be no reversal, and we are awaiting the completion of the process when we will all be reunited in the Holy Temple in Jerusalem.
The Rebbe quotes the Rambam, who says that in the first stage of the days of Moshiach, there will not be a change in the natural world order, except that the Jewish people will no longer be subjugated by the nations of the world. Our Sages and prophets desired the days of Moshiach just so that they would be able to learn Torah freely without anyone ruling over them and preventing them from doing so.
This applies to us today. Jewish people all over the world can keep Torah freely, and all the countries of the world allow the Jews to exercise their right of freedom of religion. In the light of Jewish history, this recent phenomenon is remarkable!
Finding ourselves in the first stage of the days of Moshiach, the Rebbe tells us that we have to actively await the second stage when we will see miracles that depart from the natural order.