Many nearby stars are surrounded by detectable disks of dusty material that result from the collisional erosion of planetesimals, relics of the planet formation process. Since these planetesimals persist only in stable regions like belts and resonances, the locations and physical properties of the "debris" they generate provide probes of planetary system architectures,dynamical evolution, and composition. Observations at millimeter wavelengths provide a unique window, as the large grains that dominate the emission are not much affected by stellar radiation and winds and therefore share the dynamics of the planetesimals. I will present recent results on nearby debris disks that take advantage of sensitivity and resolution now available with ALMA. In particular, I will discuss morphological features that may relate to unseen planets, surprising detections of molecular emission, and future prospects.