Quantum tunneling corresponds to the transmission of a particle with non-negligible probability through a barrier that a classical particle could not pass. This linear concept has previously been generalized to nonlinear waves and solitons incident upon an externally imposed barrier. Here, the concept of hydrodynamic tunneling is introduced whereby solitons can be transmitted through nonlinear wavetrains of hydrodynamic origin. The distinction from classical tunneling is that the barrier is not externally imposed, rather the propagation medium itself evolves and nonlinearly interacts with the soliton. The tunneling of a soliton through nonlinear wavetrains that include dispersive shock waves and Riemann (simple) waves is analyzed theoretically. The soliton transmission conditions are determined utilizing the mathematical tools of hydrodynamic theory (Riemann invariants) and nonlinear wave, Whitham modulation theory. Modulation equations for the hydrodynamic medium decouple from the modulation equation for the soliton amplitude. The transmission condition, i.e., the transmitted soliton amplitude as a function of the incident soliton amplitude and flow conditions, correspond to a Riemann invariant of the degenerate modulation equations. The soliton trajectory corresponds to a characteristic of the modulation equations. Dispersive shock waves and Riemann waves are shown to exhibit a duality principle whereby solitons tunnel through both according to the same transmission condition. The theoretical formulation is general and is applied to the Korteweg-de Vries and Nonlinear Schrodinger equations.