Chapter 5 on the method Stochastic of gradient descent is new Changes have been made to chapter 6 (the old chapter 5) on the LMS algorithm in light of the new chapter 5. The new chapter 11 on robustness is new. Minor changes have been made to finite precision effects renumbered as Chapter 12. Chapter 13 on Adaptation in Nonstationary environments , replacing the old chapter on Time varying systems, has been substantially changed to make room for new algorithms, IDBD and Autoptstep. Minor changes have been made to Kalman filters positioned as new Chapter 14. The chapter on Epilogue is brand new. The appendix on differentiation in the complex domain has been completely revised using the Wirtinger calculus. The appendix on the Langevin equation is brand new. The Bibliography is completely updated.