Were Adam and Eve real people?

The Bible clearly teaches that Adam and Eve were real people, and not just symbolic characters.  

The Apostle Paul speaks about Adam as a real person and not a symbolic figure (Romans 5:14, 1 Corinthians 15:22-45 , 1 Timothy 2:13-14), as does Jude (Jude 1:14) and the Gospel of Luke traces Jesus' genealogy back to Adam and his son Seth (Luke 3:38).   

Paul also mentions Eve in 2 Corinthians 11:3 and 1 Timothy 2:13.

Jesus Himself speaks about Abel (Matthew 23:35, Luke 11:51), as does the author of the Epistle to the Hebrews (Hebrews 11:4, 12:24), who also mentions Cain (Hebrews 11:4) as do John and Jude ( 1 John 3:12, Jude 1:11).

It's very important that Adam is a real person, because the Bible teaches that sin and death entered the world through Adam and were consequently passed on to all men:

Romans 5:12-14
Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned: For until the law sin was in the world: but sin is not imputed when there is no law. Nevertheless death reigned from Adam to Moses, even over them that had not sinned after the similitude of Adam's transgression, who is the figure of him that was to come.