We present optical colors of 72 transneptunian objects (TNOs), and infrared colors of 80 TNOs obtained with the WFPC2 and NICMOS instruments, respectively, on the Hubble Space Telescope (HST). Both optical and infrared colors are available for 32 objects that overlap between the datasets. This dataset adds an especially uniform, consistent and large contribution to the overall sample of colors, particularly in the infrared. The range of our measured colors is consistent with other colors reported in the literature at both optical and infrared wavelengths. We find generally good agreement for objects measured by both us and others; 88.1% have better than 2 sigma agreement. The median absolute magnitude, H V, magnitude of our optical sample is 7.2, modestly smaller (∼0.5 mag) than for previous samples. The median H V in our infrared sample is 6.7. We find no new correlations between color and dynamical properties (semi-major axis, eccentricity, inclination and perihelion). We do find that colors of Classical objects with i < 6° come from a different distribution than either the Resonant or excited populations in the visible at the >99.99% level with a K-S test. The same conclusion is found in the infrared at a slightly lower significance level, 99.72%. Two Haumea collision fragments with strong near infrared ice bands are easily identified with broad HST infrared filters and point to an efficient search strategy for identifying more such objects. We find evidence for variability in (19255) 1999 VK 8, 1999 OE 4, 2000 CE 105, 1998 KG 62 and 1998 WX 31.