  In this Thesis we study surfaces of general type with maximal Albanese
dimension for which the quantity $K_X^2-4\chi(\mathcal{O}_X)-4(q-2)$ vanishes
or is "small", that is surfaces close to the Severi lines. Over the complex
numbers, it is known that a surface $X$, provided that
$K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, has to satisfy the inequality
$K_X^2-4\chi(\mathcal{O}_X)-4(q-2)\geq 0$. We give a constructive and complete
classification of surfaces for which equality holds: these are surfaces whose
canonical model is a double cover of an Abelian surface ($q=2$) or of a product
elliptic surface ($q\geq 3$) branched over an ample divisor with at most
negligible singularities which intersects the elliptic fibre twice in the
latter case. We also prove, in the same hypothesis, that a surface $X$ with
$K_X^2\neq 4\chi(\mathcal{O}_X)+4(q-2)$ satisfies $K_X^2\geq
4\chi(\mathcal{O}_X)+8(q-2)$ and we give a characterization of surfaces for
which the equality holds. These are surfaces whose canonical model is a double
cover of an isotrivial smooth elliptic surface fibration branched over an ample
divisor with at most negligible singularities whose intersection with the
elliptic fibre is $4$. Because these results are intimately related to theory
of double covers, we see that their proof extend almost step by step to the
case of any algebraically closed field of characteristic different from $2$. We
also give some partial results over algebraically closed fields of
characteristic $2$ after a study of double covers in that case.
