Today's information society depends on our ability to controllably dope inorganic semiconductors, such as silicon, thereby tuning their electrical properties to application-specific demands. For optoelectronic devices, organic semiconductors, that is, conjugated polymers and molecules, have emerged as superior alternative owing to the ease of tuning their optical gap through chemical variability and their potential for low-cost, large-area processing on flexible substrates. There, the potential of molecular electrical doping for improving the performance of, for example, organic light-emitting devices or organic solar cells has only recently been established. The doping efficiency, however, remains conspicuously low, highlighting the fact that the underlying mechanisms of molecular doping in organic semiconductors are only little understood compared with their inorganic counterparts. Here, we review the broad range of phenomena observed upon molecularly doping organic semiconductors and identify two distinctly different scenarios: the pairwise formation of both organic semiconductor and dopant ions on one hand and the emergence of ground state charge transfer complexes between organic semiconductor and dopant through supramolecular hybridization of their respective frontier molecular orbitals on the other hand. Evidence for the occurrence of these two scenarios is subsequently discussed on the basis of the characteristic and strikingly different signatures of the individual species involved in the respective doping processes in a variety of spectroscopic techniques. The critical importance of a statistical view of doping, rather than a bimolecular picture, is then highlighted by employing numerical simulations, which reveal one of the main differences between inorganic and organic semiconductors to be their respective density of electronic states and the doping induced changes thereof. Engineering the density of states of doped organic semiconductors, the Fermi-Dirac occupation of which ultimately determines the doping efficiency, thus emerges as key challenge. As a first step, the formation of charge transfer complexes is identified as being detrimental to the doping efficiency, which suggests sterically shielding the functional core of dopant molecules as an additional design rule to complement the requirement of low ionization energies or high electron affinities in efficient n-type or p-type dopants, respectively. In an extended outlook, we finally argue that, to fully meet this challenge, an improved understanding is required of just how the admixture of dopant molecules to organic semiconductors does affect the density of states: compared with their inorganic counterparts, traps for charge carriers are omnipresent in organic semiconductors due to structural and chemical imperfections, and Coulomb attraction between ionized dopants and free charge carriers is typically stronger in organic semiconductors owing to their lower dielectric constant. Nevertheless, encouraging progress is being made toward developing a unifying picture that captures the entire range of doping induced phenomena, from ion-pair to complex formation, in both conjugated polymers and molecules. Once completed, such a picture will provide viable guidelines for synthetic and supramolecular chemistry that will enable further technological advances in organic and hybrid organic/inorganic devices.