Our aim is to study ordinary differential equations or simply differential systems in two real variables x— = P(x,y), (0.1) y? = Q(x,y), r 2 where P and Q are C functions defined on an open subset U of R , with— r=1,2,...,?,?.AsusualC standsforanalyticity.Weputspecialemphasis onto polynomial differential systems, i.e., on systems (0.1) where P and Q are polynomials. Instead of talking about the differential system (0.1), we frequently talk about its associated vector field—— X = P(x,y) +Q(x,y) (0.2)—x—y 2 on U? R . This will enable a coordinate-free approach, which is typical in thetheoryofdynamicalsystems.Anotherwayexpressingthevectorfieldisby writingitasX=(P,Q).Infact,wedonotdistinguishbetweenthedifferential system (0.1) and its vector field (0.2). Almost all the notions and results that we present for two-dimensional differential systems can be generalized to higher dimensions and manifolds; but our goal is not to present them in general, we want to develop all these notions and results in dimension 2. We would like this book to be a nice introduction to the qualitative theory of differential equations in the plane, providing simultaneously the major part of concepts and ideas for developing a similar theory on more general surfaces and in higher dimensions. Except in very limited cases we do not deal with bifurcations, but focus on the study of individual systems.