A comparison of X-ray images of the Sun and full disk magnetograms shows a correlation between the locations of the brightest X-ray emission and the locations of bipolar magnetic active regions. This correspondence has led to the generally accepted idea that magnetic fields play an essential role in heating the solar corona. To quantify the relationship between magnetic fields and coronal heating, the X-ray luminosity of many different active regions is compared with several global (integrated over entire active region) magnetic quantities. The X-ray measurements were made with the SXT Telescope on the Yohkoh spacecraft; magnetic measurements were made with the Haleakala Stokes Polarimeter at the University of Hawaii's Mees Solar Observatory. The combined data set consists of 333 vector magnetograms of active regions taken between 1991 and 1995; X-ray luminosities are derived from time averages of SXT full-frame desaturated (SFD) images of the given active region taken within ±4 hours of each magnetogram. Global magnetic quantities include the total unsigned magnetic flux Φtot ≡ ∫ dA|Bz|, B2z tot ≡ ∫ dAB2z, Jtot ≡ ∫dA|Jz|, and B2⊥,tot ≡ ∫ dAB2⊥, where Jz is the vertical current density and Bz and B⊥ are the vertical and horizontal magnetic field amplitudes, respectively. The X-ray luminosity LX is highly correlated with all of the global magnetic variables, but it is best correlated with the total unsigned magnetic flux Φtot. The correlation observed between LX and the other global magnetic variables can be explained entirely by the observed relationship between those variables and Φtot. In particular, no evidence is found that coronal heating is affected by the current variable Jtot once the observed relationship between LX and Φtot is accounted for. A fit between LX and Φtot yields the relationship LX 1.2 × 1026 ergs s-1(Φtot/1022 Mx)1.19. The observed X-ray luminosities are compared with the behavior predicted by several different coronal heating theories. The Alfvén wave heating model predicts a best relationship between LX and Φtot, similar to what is found, but the observed relationship implies a heating rate greater than the model can accommodate. The "Nanoflare Model" of Parker predicts a best relationship between LX and B2z,tot rather than Φtot, but the level of heating predicted by the model can still be compared to the observed data. The result is that for a widely used choice of the model parameters, the nanoflare model predicts 1.5 orders of magnitude more heating than is observed. The "Minimum Current Corona" model of Longcope predicts a qualitative variation of LX with Φtot that agrees with what is observed, but the model makes no quantitative prediction that can be tested with the data. A comparison between LX and the magnetic energy Emag in each active region leads to a timescale that is typically 1 month, or about the lifetime of an active region, placing an important observational constraint on coronal heating models. Comparing the behavior of solar active regions with nearby active stars suggests that the relationship observed between LX and Φtot may be a fundamental one that applies over a much wider range of conditions than is seen on the Sun.