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In Acts 1:8, Luke was recording the promise of the gift of the HS by Christ to the disciples and to all believers in Christ that would take place after His ascension. Jesus made a similar promise in John 7:37-39 where He said, “On the last day of the feast, the greatest day, Jesus stood up and shouted out, “If anyone is thirsty, let him come to me, and let the one who believes in me drink. Just as the scripture says, ‘From within him will flow rivers of living water.’ (Now he said this about the Spirit, whom those who believed in him were going to receive; for the Spirit had not yet been given, because Jesus was not yet glorified.)”
The meaning is that the era of the Holy Spirit had not yet arrived; the Spirit was not as yet at work in a way he later would be because Jesus had not yet returned to his Father. Cf. also Acts 19:2; Eph. 1:13-14. The HS today is God’s special gift for all those who, having realized we all have sinned and fallen short of the glory or the holy character of God, have turned from any form of self-trust or from their apathy, etc., to Jesus Christ to trust in Him by faith. This means trusting in the Bible’s witness to the person and work of Christ on the cross for our sin.
The Spirit is not anything mysterious. He is the third person of the Godhead whom the Father and the Son promised as the one to personally indwell all believers and to be their enablement to live the Christian life. This was the meaning and point in Acts 1:8, “you shall receive power … and you shall be my witnesses.”
For more on this, may we encourage you to read lessons 4 and 5 in Part Two of the ABCs for Christian Growth, Laying the Foundation. These lessons on the Holy Spirit explains something of His person and ministry with biblical evidence.