Mainly drawing on explicit examples, the author introducesthe reader to themost recent techniques to study finite andinfinite dynamical systems. Without any knowledge ofdifferential geometry or lie groups theory the student canfollow in a series of case studies the most recentdevelopments. r-matrices for Calogero-Moser systems and Todalattices are derived. Lax pairs for nontrivial infinitedimensionalsystems are constructed as limits of classicalmatrix algebras. The reader will find explanations of theapproach to integrable field theories, to spectral transformmethods and to solitons. New methods are proposed, thushelping students not only to understand establishedtechniques but also to interest them in modern research ondynamical systems.