Luke 9:27 says, "I tell you the truth, some who are standing here will not taste death before they see the kingdom of God." See also Matthew 16:28 and Mark 9:1 for the parallel quotes. In each of the synoptic Gospels, the next event immediately after this promise from Jesus is the transfiguration. Rather than interpreting Jesus’ promise as referring to His coming to establish His kingdom on earth, the context indicates that Jesus was referring to the transfiguration. The Greek word translated "kingdom" can also be translated "royal splendor," meaning that the three disciples standing there would see Christ as He really is—the King of heaven—which occurred in the transfiguration.
The "transfiguration" refers to the event described in the above cited passages when Jesus took Peter, James, and John to the top of the mountain, where He met with Moses and Elijah—representing the Law and the Prophets of the Old Testament—and spoke with them. The disciples saw Jesus in all His glory and splendor, talking with a glorified Moses and Elijah. This is a glimpse of what will occur in Jesus’ kingdom. The disciples were dumbstruck at the sight and "fell on their faces" (Matthew 17:6).
It seems most natural to interpret this promise in Matthew 16:28; Mark 9:1; and Luke 9:27 as a reference to the transfiguration, which "some" of the disciples would witness only six days later, exactly as Jesus predicted. In each Gospel, the very next passage after this promise from Jesus is the transfiguration, which shows Jesus in all His glory which will be seen again in the Kingdom of God. The contextual links make it very likely that this is the proper interpretation.