What is the biblical basis for the claim that Jesus took on the sins of mankind? It's been said that there is no biblical basis for the assumption that Jesus ever took on the sins of mankind, before, during or after his death. What is the biblical proof that Jesus did take on the sins of mankind?
There's been some really great answers here but I'm looking for some more firsthand proof, such as Matthew 5:17–18

Christ Came to Fulfill the Law
17 Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. 18 For truly, I say to you, until heaven and earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished.

Here Matthew quotes Jesus. Jesus fulfills the mosaic laws which also includes the sacrificial laws. The one perfect sacrifice that would forgive the sins of mankind past, present and future.
I believe that this Bible verse confirms that Jesus took on the sins of mankind.

John 1:29 ESV


  The next day he saw Jesus coming toward him, and said, “Behold, the
  Lamb of God, who takes away the sin of the world!