It may be that we are now entering the era of LEO constellation satellite communications after thirty years of domination by the GEO systems. This paper heralds the coming of the new era with a tutorial approach to the system design and trade‐offs of LEO constellation system design. It discusses orbital configurations, network topologies and routeing considerations, multiple access schemes and link performance design. In so doing it brings out the major design parameters and how they interact with each other. Also considered are the service applications for the LEO constellation systems, and the important difference between real‐time and delayed communication systems is highlighted. Examples of single and multi‐beam (cellular) coverage system link designs are presented for L‐Ka frequency bands. Future papers will consider aspects of the LEO spacecraft and launchers.