Measuring the density operator of a quantum systemprovides complete state information. This thesis concerns quantumstate measurement in diatomic molecules. We have developedtechniques to reconstruct the density operator of a vibrationalwave packet excited by an ultrafast laser pulse. These techniquescan be used to reconstruct the quantum state of a wave packetwithout making any assumptions about the initial state of themolecule or the amplitude and phase of the excitation pulse. Molecular emission tomography measures a phasespace quasi-probability distribution of a vibrational wave packet.We introduce tomographic state measurement, and we discuss therequirements for successful reconstruction of quantum states withcomplicated phase space distributions. Molecular tomographic state reconstructionrelies on measurements of the time-dependent spectrum (TDS) ofmolecular fluorescence. Accurate state measurement requires qualityTDS data. We describe the experimental implementation of molecularemission tomography, emphasizing the important parameters foroptimization of the TDS signal. We present phase spacequasi-probability distributions, reconstructed from experimentaldata. We introduce a new state reconstructiontechnique for directly measuring elements of the density matrix fora wave packet evolving in an anharmonic potential. We establish adirect relationship between elements of the density matrix and theTDS of molecular fluorescence. Taking advantage of thisrelationship, we analyze three methods for direct density matrixreconstruction from measurements of the TDS. The initial state of our molecular sample isdescribed by a large mixture of ro-vibrational levels. We considerthe effects of molecular rotations and a mixed initial state. Wedevelop simulations to explain the form of the TDS signals measuredin the laboratory, and we find that tomographic statereconstruction provides measurements of vibrational quantum stateswhich are relatively insensitive to large distributions of initialrotational levels. We have measured the TDS ofthe coherent component of the emission from a vibrational wavepacket. We present a model for this emission which is in excellentagreement with the measured data. We examine the effects of bothpolarization and quantum interference on the coherent signal, andwe discuss how this signal can be used for statereconstruction.