I've just written an essay on repentance and evangelism. It was very hurriedly written, but basically my point is: Unbelievers can't repent, believers must - all the time.
One of the implications is that evangelism is calling sinners to come to Christ just as they are. Two men preaching in the 19th century grasped this very well indeed.
Here is Spurgeon calling sinners to repentance:
Do not attempt to touch yourself up and make yourself something other than you really are, but come as you are to Him who justifies the ungodly. ...The Gospel will receive you into its halls if you come as a sinner, not otherwise. Wait not for reformation, but come at once for salvation. God justifieth the ungodly, and that takes you up where you now are; it meets you in your worst estate. Come in your disorder. I mean, come to your heavenly Father in all your sin and sinfulness. Come to Jesus just as you are: filthy, naked, neither fit to live nor fit to die. Come, you that are the very sweepings of creation; come, though you hardly dare to hope for anything but death. Come, though despair is brooding over you, pressing upon your bosom like a horrible nightmare. Come and ask the Lord to justify another ungodly one. (From "Justification of the Ungodly" by C.H. Spurgeon. A sermon on Romans 4:5)
And this is from a wonderful piece called Evangelical Repentance by John Colquhoun (1748-1827)
Do you postpone the act of trusting in the Lord Jesus for all His salvation, till you first sit down and mourn awhile for your sins, or till your heart be so humbled that you may be welcome to Him, and so have from your own resources a warrant for trusting in Him? Do you object against coming to Christ because you are not certain that your conviction of sin and your repentance are of the right sort? Do you apply yourself to the exercise of repentance in order to be qualified for believing in Christ, or do you apply your conscience to the commands and curses of the broken law, in order so to repent as to be entitled to trust in Him? Know, I entreat you, that this preposterous and self-righteous course will but sink you the deeper in unbelief, impenitence, and enmity to God the longer you try in this manner to seek for evangelical repentance in your heart or life, the farther you will be from finding it... Do not try to wash yourself clean in order to come to the open fountain of redeeming blood; but come to it as you are, and, by the immediate exercise of direct confidence in the Lord Jesus, wash away all your sins (Ezek 36:25).