If u is a solution to the wave equation on R^n, a local smoothing inequality bounds $\|u\|_{L^p(\mathbb{R}^n\times [1,2])}$ in terms of the Sobolev norms of the initial data. We prove Sogge's local smoothing conjecture in 2+1 dimensions. In the proof, we introduced an approximation of the $L^4$--norm that works better for induction. Another key ingredient is an incidence estimate for points and tubes. This is joint work with Larry Guth and Ruixiang Zhang.