The timescale over which molecular clouds collapse to form stars is of intense scientific interest. It reflects the support mechanisms which slow collapse such as angular momentum, thermal pressure, turbulence, and ambipolar diffusion, and impacts the chemical make up of the resulting stars and surrounding planets. In this doctoral thesis we obtain lower age limits for a set of molecular clouds by studying the ratio between atomic and molecular hydrogen gas. Clouds are initially entirely, or almost entirely composed of atomic hydrogen (HI), but by the time they form stars they are almost entirely composed of molecular hydrogen (H2 ). We measure H2 column densities using 13 CO and dust extinction as tracers. In the clouds' interiors, the HI gas is sufficiently shielded from the external radiation field that it cools to temperatures of approximately 10K to 50K, at which point it can absorb the background 21 cm emission from the warm galactic ISM. This HI Narrow Self-Absorption (HINSA) is simple to observe, yet difficult to analyze due to the complexity of the background emission. We present a novel new method of analyzing HINSA spectra which for the first time offers sufficient confidence in the derived column densities that a more sophisticated analysis of the HI/H2 ratio in clouds could be undertaken. We have performed the largest survey of HINSA in clouds carried out to date using the Green Bank Telescope (GBT), and the Five College Radio Observatory (FCRAO). We establish HINSA as a useful tool, and with the large body of data acquired, can move to determine the ages of several clouds. We discovered that the observed HINSA column densities were highly dependent on the unknown shape and orientation of each cloud. Thus we developed a geometryindependent method of determining the density distribution function of molecular clouds. A geometry-independent chemical model then allowed us to determine the lower limits to the ages of 23 clumps within 9 molecular clouds.