One of the key issues in the Reformation was the doctrine of Sola Fide, that is, that salvation is by faith alone, apart from works.
What, then, is the biblical basis for this belief?
I'm going to have to limit my quotes from Scripture and leave some compelling verses out, because there are simply so many. This answer would be even longer than my normal long-winded answers if I included all supporting Scripture references. Please forgive me if I miss something you consider relevant.
First, there is the entire book of Romans, which is a build-up to, statement of, and then supporting evidence of, the doctrine. Some relevant verses:
Chapter 1 is a description of Gentiles (non-Jews) and shows the wickedness as part of the build-up. By extension, this shows that religious affiliation (even to a so-called "one true Church" as many denominations claim) can save us.
Chapters 2 segues into showing that the Jew is no better, for even though they have the promise of God, they are still corrupt.
Chapter 3 continues to show that the Jew (and by extension everyone) is unable to attain salvation through the works of the Law and gets to verse 27:
The remainder of Romans is support for this doctrinal statement. Paul starts out by showing how the early heroes of the Jewish faith were not justified by their own goodness, but by Faith on God.
He then goes on to show how the Law is perfect, and cannot save us, but rather was given to us that we might know what sin is - so that we can realize that we are sinful.
In the words of Ray Comfort in his teaching on Hell's Best Kept Secret, God's Law is like a mirror that shows us to ourselves in our true nature. If we see we have a dirty face in a mirror, we don't wash our faces with the mirror. No, we wash our faces with water.
Likewise, we can't wash away our sins with the mirror of God's law, but instead are washed clean by the blood of Christ. This is the essence of the message to the Romans.
Other verses throughout the New Testament also state this doctrine plainly.
As I said in the beginning, a comprehensive list of Scriptures would be prohibitively large, and it would be impossible to do a better job of answering this than has already been done by Paul in Romans, but here is a link to an article with more.
In addition to David's thorough answer, I would like to add (what I think) is the single most clear verse:
(I added verse 15 for clarity, verse 16 is in bold.)
The NIV Study Bible says:
And that note is:
I think it would also be helpful to say that 'salvation by faith' is not something that only Paul talks about. He maintained that this had its bases in Old Testament scripture too when he quoted Genesis 15:6, 'And he [Abraham] believed in the LORD; and he counted it to him for righteousness' (AV/KJV). This is at a time way before the Law had been delivered to Moses, so that righteousness on the basis of the Law or good works could be discounted.
This of course becomes the basis of Paul's discussion in Romans 4:
Jesus says in many places that blindly following the Law of Moses will not lead to salvation, and is his primary criticism of practicing Jews of the time. Belief in Jesus' message and God is required for salvation.
Now, mainstream Christians take Jesus' statement "I did not come to abolish the Law but to fulfill it" as evidence that he has completed the Law and belief in him is enough going forward to attain salvation by faith alone.
However, some Christians (e.g. Messianic Jews http://en.wikipedia.org/wiki/Messianic_Jews), believe that Christians must believe in Jesus' message and God to attain salvation. But they also follow the Law of Moses as they lead their lives, not because the Law is required for salvation, but because otherwise they will be committing sin given that Jesus did not abolish the Law. Even these Christians believe in salvation by faith alone, but they do not excuse themselves from the Law believing that God will punish them for sins due to breaking the Law, before ultimately accepting them into Heaven.
There seems to be major misunderstanding of what Jesus says and preaches, and this shows in answers provided by others. Most Christians believe that Jews are wrong to rely on practicing the Law of Moses as a pathway to salvation. This is not the case. As far as I know, Judaism also believes in salvation by faith alone. The Law is to be followed to avoid displeasing God and not sin. Following the Law to the letter, without faith in the One God, is not enough.
Jews do not believe that abiding by the Law of Moses is enough for attaining salvation - belief in the One God is key to attaining salvation. Jesus saw that many Jews of his time were obsessed with details of the Law and its interpretation, and not on Moses' call to focus on the belief in the One God. Jesus came to correct them in their practice and reinforce belief in God.
I am not even close to being knowledgeable enough to answer the question in a scholarly manner, but the above difference in belief is what I have uncovered by practicing in discussions on this portal. Others may be better able to elaborate further.
In answer to the recent comment added, I will edit this answer to provide better explanation.
Since salvation is salvation from God by God and related to the state of the relationship between God and mankind, then whether one is righteous (or in right standing with God) is what is the key issue. One is saved from the punishment of God, by God, via God's own provision of the means of being in right standing with him. That is the sin atoning work of Jesus, his propitiation, and imputation of his righteousness to all those that are his and who are to accept it (God's grace).
So the above passage is simply that, the means of being called righteous is that acknowledgement of one's sinfulness and the need for God's mercy. It is simply asking God to be saved from what one deserves. Humility, repentance and acceptance that it is God who saves.
And again, the one criminal simply recognized that he was a sinner and recognized who Jesus was. By asking Jesus to remember him when Jesus enters his kingdom is a fairly profound acknowledgment that Jesus is not of this world, and that his kingdom is after death, and whatever Jewish milieu of kingdom understanding.
In both passages that I chose, someone is justified by their acknowledgement of who they are and who God is, and their relation to him.
There are countless resources providing scriptural basis for salvation by faith alone, hence why I didn't reproduce those. I chose the two passages simply because you have a clear presentation that someone standing with God is dynamically changed simply because of their acknowledgement of their relationship to God, not because they have done anything to gain favor.