We have studied electron transport through electrostatic barriers in single- and bi-layer graphene and found that the chiral nature of their quasiparticles leads to highly anisotropic tunneling, qualitatively different from the case of conventional electrons described by the Schrodinger equation. Most unusually, normally incident Dirac fermions in graphene penetrate perfectly through any barrier (similar to relativistic particles in Klein's gedanken experiment) whereas massive chiral fermions in bilayer graphene are perfectly reflected in this case. In both graphene systems, charge carriers cannot be confined in a quantum box defined by potential barriers.