We define two-person zero-sum and nonzero-sum stochastic games with finite state space and finite action space. We discuss the case of β-discounted payoffs as well as that of undiscounted or limiting time average payoffs. We look at certain classes of stochastic games that possess the orderfield property, that is, given rational inputs, such games have a rational solution. To solve such classes exactly, there is hope for finding finite arithmetic-step algorithms. We discuss algorithms for some such classes. We also briefly discuss determinacy of stochastic games with countable state space and Borel payoffs. Furthermore, we outline some applications of stochastic games as well as some recent work.