The properties of molecular aggregates, coupled clusters of small molecules, are often challenging to unravel because of their inherent complexity and disordered environments. Their structure-function relationships are often far from obvious. However, their ability to efficiently channel excitation energy over remarkable distances, as is the case in photosynthetic light harvesting, is a compelling motivation to investigate them. Understanding and subsequently mimicking the processes in photosynthesis, for example, will set the stage for considerable advances in using light harvesting to fuel renewable energy technologies. Two-dimensional (2D) electronic spectroscopy is emerging as a nonlinear optical technique that provides significant insight into the interactions and dynamics of complex molecular systems. In addition to spectrally resolving excitation and emission energies over significant bandwidths with femtosecond resolution, this technique has already enabled discoveries about the structure and dynamics of photosynthetic light-harvesting complexes and other aggregates. Multiple capabilities unique to 2D electronic spectroscopy enable such findings. For example, the spectral resolution of excitation and emission combined with the ability to eliminate the effects of static disorder can reveal the homogeneous line width of a transition and the different dynamic contributions to it. Two dimensional spectroscopy is also sensitive to electronic coherence and has been employed to identify and characterize coherent excitation energy transfer dynamics in photosynthetic systems and conjugated polymers. The presence of cross-peaks, signals for which excitation and emission occur at different wavelengths, provides multiple forms of information. First, it allows the identification of states in congested spectra and reveals correlations between them. Second, we can track excitation energy flow from origin to terminus through multiple channels simultaneously. Finally, 2D electronic spectroscopy is uniquely sensitive to intermolecular electronic coupling through the sign and amplitude of the cross-peaks. This feature makes it possible to reveal spatial molecular configurations by probing electronic transitions. Another means of "resolving" these angstrom-scale arrangements is to manipulate the probing laser pulse polarizations. In this way, we can isolate and modulate specific processes in order to retrieve structural information. In this Account, we demonstrate these capabilities through a close collaboration between experiments and modeling on isolated photosynthetic pigment-protein complexes and also on J-aggregates. Each of the probed systems we describe offers insights that have both increased the utility of 2D electronic spectroscopy and led to discoveries about the molecular aggregates' dynamics and underlying structure.