Paul announces that Jesus Christ is our hope (1 Tim 1:1). Parallel to those passages in the Old Testament where those who hope are not put to shame, Paul says hope does not disappoint us (Rom 5:5). The reason is that we already have a taste of the future glory because of the love with which the Holy Spirit fills our hearts. In other words, the gifts of love and of the Spirit are downpayments of future glory for which we hope (Rom 5:2; cf. Eph 1:13-14).
In the Old Testament hope has to do with waiting for, looking for, desiring. This is paralleled in the Gospels, where the word "hope" is not very frequent but the idea of looking expectantly is. Simeon looked for Israel's consolation at the advent of the Messiah (Luke 2:25-26). Likewise, Anna, the prophetess, upon recognizing who Jesus was, proclaimed him to all those who were anticipating redemption (Luke 2:36-38).
In connection with hope in Romans 8:18-25 Paul speaks of waiting with eager expectation for the revelation of the children of God (v. 19), waiting for the adoption as sons (v. 23). We are waiting "for the righteousness for which we hope" (Gal 5:5) and for "the blessed hope, " namely, the glorious appearing of our Lord (Titus 2:13). Paul has both an eager expectation and a hope for God to be glorified in him, whether in life or death (Php 1:20). He goes on to express his desire to leave this world to be present with Christ (1:23).
As hope is connected with patient endurance in the Old Testament, so in the New Testament trials lead to hope (Rom 5:3-4) and hope is steadfast (1 Th 1:3). When we hope for something we wait for it through patience (Rom 8:25; 15:4). ...New Testament hope is primarily eschatological. After being introduced late in Old Testament times, hope in the resurrection of the dead grew in the intertestamental period in such proportion that Paul could speak of the resurrection as the "hope of Israel" (Acts 28:20; 24:15; 26:6-8). - Baker's Evangelical Dictionary of Biblical Theology