Suppose that k is a field and that 2g-2+n > 0. Then one has the moduli space M_{g,n} of n pointed smooth projective curves of genus g over k. The generic curve of type (g,n) over k is defined to be the restriction of the universal curve over M_{g,n} to its k-generic point Spec k(M_{g,n}). This curve has n tautological rational points. Results in Teichmuller theory imply that when g > 2 and char(k)= 0, these are the only rational points. Recent results of Watanabe imply that this also holds when char(k)>0. When n = 0, results of Hain in char 0 and Watanable in positive characteristic imply that Grothendieck´s Section Conjecture holds for the generic curve of genus g > 2. This talk will survey these results and introduce the tool of weighted completion that was used to prove them.