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<jats:title>ABSTRACT</jats:title> <jats:p>The formation timescale and final architecture of exoplanetary systems are closely related to the properties of the molecular disks from which they form. Observations of the spatial distribution and lifetime of the molecular gas at planet-forming radii (<jats:italic>a</jats:italic> &lt; 10 AU) are important for understanding the formation and evolution of exoplanetary systems. Toward this end, we present the largest spectrally resolved survey of H<jats:sub>2</jats:sub> emission around low-mass pre-main-sequence stars compiled to date. We use a combination of new and archival far-ultraviolet spectra from the Cosmic Origins Spectrograph and Space Telescope Imaging Spectrograph instruments on the <jats:italic>Hubble Space Telescope</jats:italic> to sample 34 T Tauri stars (27 actively accreting Classical T Tauri Stars and 7 non-accreting Weak-lined T Tauri Stars) with ages ranging from ∼1 to 10 Myr. We observe fluorescent H<jats:sub>2</jats:sub> emission, excited by Lyα photons, in 100% of the accreting sources, including all of the transitional disks in our sample (CS Cha, DM Tau, GM Aur, UX Tau A, LkCa 15, HD 135344B, and TW Hya). The spatial distribution of the emitting gas is inferred from spectrally resolved H<jats:sub>2</jats:sub> line profiles. Some of the emitting gas is produced in outflowing material, but the majority of H<jats:sub>2</jats:sub> emission appears to originate in a rotating disk. For the disk-dominated targets, the H<jats:sub>2</jats:sub> emission originates predominately at <jats:italic>a</jats:italic> ≲ 3 AU. The emission line widths and inner molecular radii are found to be roughly consistent with those measured from mid-IR CO spectra.</jats:p>