We typically think of quantum physics as applying to the motion of atom-scale objects. Advances in atomic and device physics are now letting us access quantum phenomena in the collective motion of objects composed of thousands to billions of atoms. In this talk, I will discuss recent results from our experiment at Berkeley, in which we use the collective center-of-mass motion of a cloud of ultracold atoms as a many-atom mechanical oscillator, measured with light in a high-finesse optical cavity. We have observed new quantum phenomena: the generation of a nonclassical state of light arising from optical feedback with a massive moving object, as well as a direct measurement of the quantization of a many-atom mechanical oscillator. I will also discuss how ideas from optomechanics can be applied to collective magnets composed of large numbers of atomic spins, including how strong cavity-spin coupling can lead to a many-body quasiparticle, and how cavity-mediated interactions might be used to simulate quantum magnetism.