In addition to its spectrum and temperature anisotropy, the 2.7 K cosmic microwave background (CMB) is also expected to exhibit a low level of polarization. The spatial power spectrum of the polarization can provide details about the formation of structure in the universe as well as its ionization history. Here we calculate the magnitude of the CMB polarization in various cosmological scenarios, with both an analytic and a numerical method. We then outline the fundamental challenges to measuring these signals and focus on two of them: achieving adequate sensitivity and removing contamination due to foreground sources. We describe the design of a ground-based instrument (Polarization Observations of Large Angular Regions) that could detect polarization of the CMB at large angular scales in the next few years.