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I will begin with a brief overview of some well-known results in incidence geometry and go on to discuss a recent joint work with Thang Pham and Audie Warren, in which we prove upper bounds on the number of incidences between sets of points and conics over finite fields. I will conclude the talk by considering applications to certain finite field variants of Erd\H{o}s type problems on the number of distinct algebraic distances formed by point sets, including improvements to results of Koh and Sun (2014) and Shparlinski (2006).