This series of lectures aims to introduce symbolic powers of ideals and geometric reasons for studying them. The first lecture will present the early origins of symbolic powers in commutative algebra and the geometric characterization for symbolic powers of radical ideals as higher order vanishing ideals due to Zariski and Nagata. The second lecture will survey 21st century developments in the study of symbolic powers, with a focus on the containment problem between the symbolic and ordinary powers of an ideal. The final lecture will discuss higher order polynomial interpolation problems in algebraic geometry, some asymptotic invariants associated to them, and their connection to symbolic power ideals.