Could human civilisation spread across the whole Galaxy?

Could human civilization eventually spread across the whole Milky Way Galaxy? Could we move beyond our planet to establish colonies in the multitude of star systems out there? These questions are pretty huge, but their (theoretical) answers were actually put forward decades ago. Roey Tzezana describes the conceptual von Neumann machine. 

https://youtu.be/Rpy9Qp7NAaw