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Abstract: In these two talks, we introduce the unique continuation for elliptic and parabolic equations. In the first talk, we will discuss the strong and weak unique continuation for elliptic equations. We will mention some open problems, exciting progress made by Logunov, et. al., and a proof of a simple unique continuation result using the Carleman estimate. In the second talk, we will discuss the application of a parabolic unique continuation result in the regularity criteria for 3D Navier-Stokes equations. The second talk will be based on Escauriaza-Seregin-Sverak's 2003 ARMA paper.