In the past (ninteen-eightees), it was best to use a standard processors as a separate chip.
Today, it is no problem to put down one or more 'standard' processors on a SoC.
It is also quite easy to design your own, so MIPS, Tensilica, ARM and other CPU core providers have to compete against in-house design teams.
For instance, we use the the totally free OR 1000 in the practical materials of this course.