These notes are centred on one question : given a real algebraic surface X determine the topology of the real part X(R). Of course, since, to quote Hartshorne, the guiding problem in algebraic geometry is the classification problem (and this goes for real algebraic geometry also), the latter is very present in these notes. In fact it is present to the point, that we have only obtained a precise answer to our original question when we have obtained a precise answer to the classification problem. In this sense one could say that the underlying theme (and even, the main theme) of these notes is the classification problem of real algebraic surfaces.
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