This fullness of the Spirit was not to remain uniquely the Messiah's, but was to be communicated to the whole messianic people. On several occasions Christ promised this outpouring of the Spirit, a promise which he fulfilled first on Easter Sunday and then more strikingly at Pentecost. Filled with the Holy Spirit the apostles began to proclaim "the mighty works of God," and Peter declared this outpouring of the Spirit to be the sign of the messianic age. Those who believed in the apostolic preaching and were baptized received the gift of the Holy Spirit in their turn.
Source: Catechism of the Catholic Church (Vatican.va)