Quantitatively measuring the mechanical properties of soft matter over a wide range of length and time scales, especially if a sample is as complex as typical biological materials, remains challenging. Living cells present a further complication because forces are generated within these nonequilibrium materials that can change material properties. We have here developed high-bandwidth techniques for active one- and two-particle microrheology to tackle these issues. By combining active micromanipulation of probe particles with an optical trap with high-resolution tracking of thermal motions of the very same particles by laser interferometry, we can both measure the mechanical properties of and, at the same time, identify nonequilibrium forces in soft materials. In both simple liquids and equilibrium cytoskeletal actin networks, active microrheology (AMR) proves to be less noise sensitive than and offers extended bandwidth (0.1-100 kHz) compared to passive microrheology (PMR), which merely tracks thermal motions. We confirm high-frequency power-law dynamics in equilibrium actin networks with two-particle AMR and also discuss low-frequency local mechanical response near probe particles which shows up in one-particle AMR. The combination of AMR and PMR allowed us to quantify nonthermal force fluctuations in actin networks driven by myosin motor proteins. Our approach offers a new direct way to investigate the nonequilibrium dynamics of living materials.