The Geometric Langlands Conjecture (GLC) for a curve C and a group G is a non-abelian generalization of the relation between a curve and its Jacobian. It claims the existence of Hecke eigensheaves on the moduli of G-bundles on C. The parabolic GLC is a further extension to curves with punctures. After explaining and illustrating the conjectures, I will outline an approach to proving them using non-abelian Hodge theory. A key geometric ingredient is the locus of wobbly bundles: bundles that are stable but not very stable. If time allows, I will discuss two instances where this program has been implemented recently: GLC for G=GL(2) and genus 2 curves (with T. Pantev and C. Simson), and parabolic GLC for P1 with marked points (with T. Pantev).