In this paper, we survey techniques for routing and performance analysis of Low Earth Orbit (LEO) systems. We first review the basic structure of LEO satellite communications and present the main characteristics of LEO systems. The dynamic characteristics of LEO systems make the performance analysis and routing problem quite complicated. In this study, we review some of the proposed solutions to these problems. We explain the two main criteria for the calculation of performance analysis: blocking probability of ongoing calls during handover and the new call blocking probability. Then we mention which way is the most preferable and what are done during the calculation process.