This question has many different answers for every part of the world. Paleolithic, let's say hunter gatherer societies, which is nomad and depended on natural resources, lived up to 4-5 thousand years ago.
We can define Neolithic as village life and agriculture.
First Neolithic villages are found in southeastern Anatolia and northern Syria and Iraq. All these first villages dated back to 10-12 thousand years ago. In central Anatolia there are some Neolithic sites too which are dated to the same period. There is no use of pottery at that time, for that reason it's called "pre-pottery Neolithic" by prehistorians.
After the "collapse" of pre-pottery societies we see a migration movement out of this "core" region of village societies. This wave of migration reached western Anatolia and the Balkans 7-8 thousand years ago and reached western Europe nearly 4 thousand years ago.
In Japan, Paleolithic people settled in the Paleolithic. This is called "Jeitun culture" (24 thousand years ago). But when they actually became Neolithic they returned to nomad life.
In Americas, people started to settle 5-6 thousand years ago. As far as I know, it started in the Central America.
But we must keep in mind that when some people started to settle, there had been a lot of people who rejected this lifestyle and stayed Paleolithic.
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