We prove two types of inequalities for a foliation of general type on a smooth projective surface, the slope inequality and Noether inequality, both of which provide lower bounds on the volume \(\mathrm{vol}(\mathcal F)\). In order to define the slope, we first introduce three birational non-negative invariants \(c_1^2(\mathcal F)\), \(c_2(\mathcal F)\) and \(\chi(\mathcal F)\) for any foliation \(\mathcal F\), called the Chern numbers. If the foliation \(\mathcal F\) is not of general type, the first Chern number \(c_1^2(\mathcal F)=0\), and \(c_2(\mathcal F)=\chi(\mathcal F)=0\) except when \(\mathcal F\) is induced by a non-isotrivial fibration of genus \(g=1\). If \(\mathcal F\) is of general type, we obtain a slope inequality when \(\mathcal F\) is algebraically integrable, which gives a lower bound on \(\mathrm{vol}(\sF)\) by \(\chi(\mathcal F)\). On the other hand, we also prove three sharp Noether type inequalities for a foliation of general type, which provides a lower bound on \(\mathrm{vol}(\mathcal F)\) by the geometric genus \(p_g(\mathcal F)\). As applications, we also give partial solutions to the Poincar\'e and Painlev\'e problems using these two inequalities. This is a joint work with Professor S.L. Tan.