The two-person differential simple pursuit game on the sphere is considered and the following problems are discussed. What is the fastest strategy for the pursuer? What does the Appolonius circle on the sphere look like? Is there universal pursuit strategy on the sphere? Can the evader avoid the collision with the pursuer? The well-known pursuit Pi-strategy for the same game on the plane is extended to the sphere by two different ways. A variant of the extension is discussed, the differences between games on the plane and on the sphere are discovered, and the proofs of the properties of the strategy on the sphere are obtained.