In his famous Treatise on True Devotion to Mary, St Louis Marie Grignion de Montfort admirably describes the expectation over the coming of such a reign — the “century of Mary” or “Reign of Mary.”
‘The great apostle of devotion to Our Lady says:
“"When will that happy day come," asks a saintly man of our own day whose life was completely wrapped up in Mary, "when God's Mother is enthroned in men's hearts as Queen, subjecting them to the dominion of her great and princely Son? When will souls breathe Mary as the body breathes air?" When that time comes wonderful things will happen on earth. The Holy Spirit, finding his dear Spouse present again in souls, will come down into them with great power.
“He will fill them with his gifts, especially wisdom, by which they will produce wonders of grace. My dear friend, when will that happy time come, that age of Mary, when many souls, chosen by Mary and given her by the most High God, will hide themselves completely in the depths of her soul, becoming living copies of her, loving and glorifying Jesus? That day will dawn only when the devotion I teach is understood and put into practice. Ut adveniat regnum tuum, adveniat regnum Mariae: “Lord, that your kingdom may come, may the reign of Mary come!”’ 
‘That your kingdom may come, may the reign of Mary come — that is, the reign of Jesus Christ. This is the thought that permeates the whole Treatise on True Devotion, and which is expressed here with unparalleled clarity and ardor.
‘The devotion St. Louis Marie Grignion de Montfort preaches is slavery of love to the Blessed Mother, the theme expounded in the Treatise.
‘We identify this century of Mary, or Reign of Mary, of which the Saint speaks with the triumph of the Immaculate Heart of Mary that Our Lady announced at Fatima; and this illation is obvious.
‘The Fatima Message clearly says today’s world is laden with crimes but a day will come when the Immaculate Heart of Mary will triumph. Evidently, this should be understood in the sense that the Blessed Mother will establish her empire upon souls and therefore upon institutions, nations, and upon the whole world.
‘Hence, from the Fatima standpoint, the Reign of Mary will be specifically the Reign of the Immaculate Heart of Mary. That is, a reign of the purity and goodness of the Mother of God’s maternal heart; a reign of great splendor both in temporal society and in the Church through abundant graces poured by the Holy Ghost.’ 
(Taken from informal lectures of Prof. Plinio Correa de Oliveira, and have not be revised by the author.)
 Treatise on True Devotion to the Blessed Virgin, no. 217.
 Plinio Corrêa de Oliveira, Guerreiros da Virgem — A Réplica da Autenticidade, Ed. Vera Cruz, São Paulo, 1985, p. 228.