Emission and polarization by Galactic dust is an important Galactic foreground that contaminates cosmic microwave background (CMB) signal. In the first part of the talk, I will talk about our improved model of microwave emission arising from rapidly spinning small dust grains, which was proposed to explain anomalous emission discovered about 20 years ago. The second part is devoted to dust polarization that severely interferes with CMB polarization experiments to study the early Universe. I will first discuss recent progress in understanding the alignment of interstellar dust grains with magnetic fields and modeling of polarization by aligned grains, which includes an analytical and predictive model of alignment based on radiative torques proposed by our group. Then, I will present theoretical predictions and observational constraints for the polarization of spinning dust emission.