A long-standing problem in low-mass star formation is the "luminosity problem," whereby protostars are underluminous compared to the accretion luminosity expected both from theoretical collapse calculations and arguments based on the minimum accretion rate necessary to form a star within the embedded phase duration. In this dissertation, I present new research on protostars and the protostellar accretion process that addresses the luminosity problem in the following ways: I report new infrared detections of a very low luminosity protostar in Taurus and use all existing data ranging from the infrared through millimeter wavelengths to constrain radiative transfer models and determine physical properties of the source. I argue that the derived source luminosity is lower than that expected based on the properties of a previously detected molecular outflow driven by this source and suggest that this discrepancy can be resolved by variable rather than constant mass accretion. I report the discovery of a new protostar that is also driving a molecular outflow. Following a similar modeling procedure as above, I show that this source has an even lower luminosity that is once again inconsistent with that expected based on the properties of its outflow, again suggesting variable mass accretion. I present the results of a complete search for all protostars with luminosities less than or equal to that of our Sun in a new infrared survey of nearby star-forming regions. I identify 50 protostars with such luminosities. Only a small fraction (15-25%) of dense cores thought to be starless (not yet collapsing to form stars) in fact harbor low luminosity protostars. The distribution of luminosities of these 50 protostars is inconsistent with a constant protostellar mass accretion rate. I present a set of evolutionary models that start with existing models following the inside-out collapse of singular isothermal spheres and add isotropic scattering off dust grains, a circumstellar disk, two-dimensional envelope structure, mass-loss and the opening of outflow cavities, and a simple treatment of episodic mass accretion. I conclude that episodic mass accretion is both necessary and sufficient to resolve the luminosity problem. / text