How will we recognize dark matter when we see it? And what kinds of particle properties may it have? In this talk, I will discuss two aspects of finding and characterizing dark matter. First, I will discuss a new signal for WIMP dark matter: gravitational focusing in direct-detection experiments. This effect leads to an energy-dependent phase-shift in the peak direct-detection event rate throughout the year. I will discuss this in light of current putative annual-modulation claims. Second, I will talk about strong dark-matter self-interactions and astronomical observations. I will show that large cross sections may be lurking, and how different types of self-interactions lead to different dark-matter interpretations of observations of matter distributions in the sky.