Concise Bible Dictionary:
The word translated “world” in the following passages is αἰών (see WORLD): it may therefore be rendered “the age to come.” The Lord declared that the blasphemy against the Holy Ghost of attributing His miracles to the power of Satan, should not be forgiven in the present age, nor in the age to come (Matt. 12:32). He also promised that those who gave up property or earthly relations for the sake of the kingdom of God, should, beside present blessing, have eternal life in the age to come (Mark 10:30; Luke 18:30). Christ has been exalted above every name named in this world or in the world to come (Eph. 1:21). In Hebrews 6:5 Paul speaks of some who had tasted of the powers of the age to come, doubtless alluding to miracles.
The word in Hebrews 2:5 is οἰκουμένη, “habitable earth.” The habitable earth to come is not put in subjection to angels, but to the Son of Man.
The “present age” was well understood by the Jews to be in contrast to that age which should be introduced by the Messiah. Saints in Old Testament times looked forward to this, as for example Abraham (John 8:56; Rom. 4:13; Heb. 11:10). Christians are able by faith to enjoy even now the blessings of that coming age, when the Lord with universal sway will establish His kingdom on earth, and have everything under His administration (Compare Rom. 14:17). Every moral question will be settled then. The world to come will usher in eternity (1. Cor. 15:24,28).
From Anstey’s Doctrinal Definitions:
This expression refers to the millennial kingdom of Christ (Matt. 12:32; Mark 10:30; Luke 18:30; Eph. 1:21; Heb. 6:5).
There are at least ten different expressions that refer to the time when Christ will have His day of glory and power in this world:
• The World to Come (Matt. 12:32; Mark 10:30; Eph. 1:21; Heb. 6:5).
• The Kingdom of the Heavens (Matt. 4:17; 6:10; 7:21; 8:11; 10:7).
• The Millennium (Rev. 20:6 – in the Latin Vulgate version).
• The Times of Refreshing (Acts 3:19).
• The Restitution of All Things (Acts 3:21).
• The Regeneration (Matt. 19:28).
• The Reformation (Heb. 9:10).
• The Dispensation of the Fulness of Times (Eph. 1:10).
• The Day of the Lord (1 Thess. 5:2; 2 Thess. 2:2).
• The Day of Christ (1 Cor. 1:8; 3:13; 5:5; 2 Cor. 1:14; Phil. 1:6, 10; 2:16).