Robust subspace tracking (RST) can be simply understood as a dynamic (time-varying) extension of robust PCA. More precisely, it is the problem of tracking data lying in a fixed or slowly-changing low-dimensional subspace while being robust to sparse outliers. This work develops a recursive projected compressive sensing algorithm called ``Nearly Optimal RST (NORST)'', and obtains one of the first guarantees for it. We show that NORST provably solves RST under weakened standard RPCA assumptions, slow subspace change, and a lower bound on (most) outlier magnitudes. Our guarantee shows that (i) NORST is online (after initialization) and enjoys near-optimal values of tracking delay, lower bound on required delay between subspace change times, and of memory complexity; and (ii) it has a significantly improved worst-case outlier tolerance compared with all previous robust PCA or RST methods without requiring any model on how the outlier support is generated.