In the first two and a half lectures I will thouch these themes:

1. generalities on divisors and linear systems on a surface;
2. (birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);
3. generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.
4. I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.