The phenomenon of resonant tunneling transport through polar double-barrier heterostructures is systematically investigated by a combined experimental and theoretical approach. On the experimental side, GaN/AlN resonant tunneling diodes (RTDs) are grown by molecular beam epitaxy. In situ electron diffraction is used to monitor the number of monolayers incorporated into each tunneling barrier of the RTD active region. Using this precise epitaxial control at the monolayer level, we demonstrate exponential modulation of the resonant tunneling current density as a function of barrier thickness. At the same time, both the peak voltage and the characteristic threshold bias exhibit a dependence on barrier thickness as a result of the intense electric fields present within the polar heterostructures. To get further insight into the asymmetric tunneling injection originating from the polar active region, we present an analytical theory for tunneling transport across polar heterostructures. A general expression for the resonant tunneling current that includes contributions from coherent and sequential tunneling processes is introduced. After the application of this theory to the case of GaN/AlN RTDs, their experimental current-voltage characteristics are reproduced over both bias polarities, with tunneling currents spanning several orders of magnitude. This agreement allows us to elucidate the effect of the internal polarization fields on the magnitude of the tunneling current and broadening of the resonant tunneling line shape. Under reverse bias, we identify new tunneling features originating from highly attenuated resonant tunneling phenomena, which are completely captured by our model. The analytical form of our quantum transport model provides a simple expression that reveals the connection between the design parameters of a general polar RTD and its current-voltage characteristics. This new theory paves the way for the design of polar resonant tunneling devices exhibiting efficient resonant current injection and enhanced tunneling dynamics as required in various practical applications.