In this chapter, we begin making a brief review on the history of the studies of planet Mars. Then, we review the modern research (with recent status, data, results, and progress) on the search for possible extinct or extant life on Mars, with special emphasis on the search for the presence (in the past and/or presently) of liquid water within Mars' surface and subsurface -a prerequisite for the evolution from geochemical state to biogeochemical state, as we are aware of here on planet Earth. Through the chapter, I present some proposals of mine about the astrobiology of Mars. We also analyze recent astrobiological experiments on board the International Space Station (ISS) for the future exploration of Mars. And we make a brief review on the evolution of equipment for its exploration and of future manned presence on the beautiful planet Mars -the "Red Planet."