Basileuō means to reign, rule, or be king — the exercise of royal authority. Appearing about 21 times in the NT, it is the verbal form of basileus (king). It describes past human kingdoms (the reign of sin and death), the present reign of grace and Christ, and the future eternal kingdom. The verb captures the dynamic, active exercise of sovereignty.
Basileuō structures Paul's entire argument in Romans 5:12–21 — a battle of reigns. Sin reigned in death (ebasileosen ho thanatos), but grace now reigns through righteousness (basileuein hē charis). The transfer of reign is the essence of the gospel: from Adam's death-reign to Christ's life-reign. 1 Corinthians 15:25 declares "he must reign (basileuein) until he has put all his enemies under his feet" — an active, progressive sovereignty. Revelation celebrates the moment when "the Lord our God Almighty reigns [ebasileuses]" (Revelation 19:6) — the completion of the kingdom.