A criminal being crucified at the same time as Jesus Christ said to Him, "Lord, remember me when You come into Your kingdom" (Luke 23:42). Jesus' response is recorded in the New King James Version as, "Assuredly, I say to you, today you will be with Me in Paradise" (Luke 23:43).
When Christ died later that day, did He and the criminal go to heaven? The answer lies in John 20:17, where Christ says to Mary Magdalene (three days after His crucifixion), "I have not yet ascended to My Father." So Christ hadn't returned to heaven the day of His crucifixion.
The system of punctuation we are used to in the English language—which includes quotation marks, commas, semicolons, etc.—was not used in the Greek language that the Gospel of Luke was written in. As a result, English translators of the Bible had to place punctuation into the Bible based on the context of the translation. While they have generally done an accurate job with this, their lack of understanding of the timing of the resurrections and their belief in nonbiblical doctrines such as the immortality of the soul led them to incorrectly punctuate Luke 23:43.
So what did Jesus mean? A common phrase of Jesus' day was the emphatic "I say to you today," with which Jesus prefaced His statement that "you will be with Me in Paradise." Christ wasn't saying, "You will be with Me in Paradise today," but rather, "Today I'm telling you, you will be with Me in Paradise."
For further treatment of the subject, including an explanation of the connection between paradise and the Kingdom of God that the thief asked about, please read our booklets Heaven and Hell: What Does the Bible Really Teach? and Jesus Christ: The Real Story.