The elastic properties of silica phases are reviewed. Available monocrystal data for crystalline SiO2 polymorphs (low-quartz, high-quartz, low-cristobalite, high-cristobalite, stishovite) are collected from the literature, and effective elastic constants (Young's moduli, shear moduli, bulk moduli and Poisson ratios) are calculated from these using Voigt-Reuss-Hill averaging. Both experimental data and simulation results are taken into account. A table of room temperature elastic constants for crystalline silica polymorphs and silica glass is given that lists the recommended current "state-of-the-art" values. All data are consistent with the well-known auxetic behavior of cristobalite at room temperature, and high-temperature simulation data published for cristobalite confirm aztxetic behavior for all temperatures from room temperature up to more than 1500 degrees C. The calculations of this paper show that also quartz can be auxetic, but only in a very limited temperature range around the low-to-high-quartz transition temperature (420 - 577 degrees C). Experimental measurements of elastic properties of tridymite and cristobalite, including high-temperature measurements, are identified as a desideratum of future research.