Differential algebraic groups were introduced by P. Cassidyand E. Kolchin and are, roughly speaking, groups defined byalgebraic differential equations in the same way asalgebraic groups are groups defined by algebraic equations. The aim of the book is two-fold: 1) the provide an algebraicgeometer's introduction to differential algebraic groups and2) to provide a structure and classification theory for thefinite dimensional ones. The main idea of the approach is torelate this topic to the study of: a) deformations of (notnecessarily linear) algebraic groups and b) deformations oftheir automorphisms. The reader is assumed to possesssomestandard knowledge of algebraic geometry but no familiaritywith Kolchin's work is necessary. The book is both aresearch monograph and an introduction to a new topic andthus will be of interest to a wide audience ranging fromresearchers to graduate students.