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There has been growing interest in recent years on random objects in algebraic geometry. The expected number of real roots of a univariate polynomial has been studied for different probability measures on the space of polynomials, by many authors. After discussing some of these results, I will switch to multivariate polynomials and survey some of the known results regarding the expected number of connected components of a real algebraic plane curve and their expected volumes. Finally, I will present some of our recent results with Turgay Bayraktar regarding the expected depth of a real algebraic plane curve.