Here's the New Testament reference to the "Last Days". It's from Acts Chapter 2: v.15
For these are not drunken, as ye suppose, seeing it is but the third hour of the day. But this is that which was spoken by the prophet Joel; And it shall come to pass in the last days, saith God, I will pour out of my Spirit upon all flesh: and your sons and your daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams:
Apparently we've been in the "Last Days" since Peter stood up and spoke at Pentecost. There's lots more info here: Prohpecy Reformation Institute