In this paper, the basic aspects of optimum and adaptive filters are introduced and classical LMS, RLS methods are reviewed along with their many variants. In addition to convergence behavior and performance of adaptive filters, their processing complexity for real-time implementations are also compared. Both FIR and IIR adaptive filters and their practical applications are described in detail. Recent developments including total least mean squares (TLMS) based adaptation and blind methods are briefly discussed. After building up the essential theoretical framework, this paper will devote more time to real-time implementation issues and some key applications such as: echo cancellation for voice and data; equalization; cross-talk mitigation; carrier recovery in presence of phase jitter; and bio-medical applications.