"He also said, 'Blessed be the Lord, the God of Shem! May Canaan be the slave of Shem.'" NIV translation
We notice that Noah said "the God of Shem" as he was the oldest and so, through him, would come Abraham (the father of Israel). The removal of God's protection from Canaan was fulfilled when Israel (descendants of Shem) left Egypt and took possession of the land of Canaan. This is also a picture of the millennial reign where God's people will reign on the earth for a thousand years (see Revelation 20).
Now, we see the future of Japheth whose name, in Hebrew, means "extend".
In him, we see a picture of the Gentiles united to live with the Jews and rule the world (which will happen during the millennial reign).
"After the flood Noah lived 350 years. Altogether, Noah lived 950 years, and then he died." NIV translation
We see that there was a period of 350 years where Noah and his sons increased in number on the earth as they were told. Although we do not know how many people there were on the earth after this, we know that we are all descendants of Noah through one of these sons.