What is the purpose of a sheriff's office?

Sheriifs serve and protect a county. Rural areas are big and spread out and so dont have the money to afford there own city Police. Also there is a lot of land that people visit and occupy that dont fall within city limits. One example is in the deserts of California and Arizona. During the summer tens of thousands of people flock to the dunes in the middle of no where. Without a sheriff there would be no police force that had juristiction to keep the peace and promote safety.