The object of this book is the presentation, analysis and extension of the Forward Improvement Iteration Algorithm (FII) published 1980 by Albrecht Irle and similar algorithms published 2006 and 2008 by Christian Bender, Anastasia Kolodko, and John Schoenmakers and published 2010 by E. L. Presman, as well as giving adequate applications. The FII algorithm solves problems of optimal stopping in discrete time by giving a lower approximation of the optimal stopping time and a lower bound of the Snell envelope in each iteration step, which converge to the Snell envelope resp. optimal stopping time. The similar algorithms mentioned above introduced additional parameters for the special cases they treated. This book shows that these algorithms can be put together into a single extended FII algorithm, inheriting the additional parameters and results. Furthermore the algorithm is transfered to the Markov case, also inheriting reasonable additional parameters. Some numerical applications and an investigation of the monotone case and some best choice problem, for which already known solutions of some special cases are extended to a general solution, are also given.