The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map f: [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] → X to the surface is constant. In 1970, Kobayashi conjectured more generally that a (very) generic hypersurface of sufficiently high degree in projective space is hyperbolic. Our technique follows the stream of ideas initiated by Green and Griffiths in 1979, which consists of considering jet differentials and their associated base loci. However, a key ingredient is the use of a different kind of jet bundle, namely the "Semple jet bundles" previously studied by the first named author. The base locus calculation is achieved through a sequence of Riemann-Roch formulas combined with a suitable generic vanishing theorem for order 2-jets. Our method covers the case of surfaces of general type with Picard group [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] and (13 + 12θ2)c21 -9c2 > 0, where θ2 is the "2-jet threshold" (bounded below by -1/6 for surfaces in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]3). The final conclusion is obtained by using recent results of McQuillan on holomorphic foliations.