The new framework of random polynomials developed by R. Pemantle, I. Rivin and the late O. Schramm has been studied in this thesis. The strong Pemantle-Rivin conjecture asks whether for random polynomials with independent and identically distributed zeros with a common probability distribution μon the complex plane, the empirical measures of their critical points would converge weakly to μ almost surely. This convergence question has connection with geometry of polynomials. S. D. Subramanian confirmed the conjecture whenμis a non-uniform distribution supported in the unit circle ∂D. In this thesis, the conjecture has been extended to considering not only the critical points of the random polynomials, but also the zeros of their higher order, polar and Sz.-Nagy's generalized derivatives. The case thatμ(uniform or not) is supported in ∂D has been studied, where the derivative of each order has been proved to satisfy the conjecture. Subramanian's work has thereby been completed plus generalization. The same almost sure weak convergence has also been shown for polar and Sz.-Nagy's generalized derivatives, under some mild conditions. In particular, the result on polar derivative is the crux of filling up Subramanian's missing case of uniform μ. Meanwhile, the original Pemantle{Rivin conjecture asks about convergence in probability instead of the aforesaid stronger almost sure convergence. Z. Kabluchko fully solved this conjecture. In this thesis, his methodologies have been adapted to prove the analogous conjecture for random finite Blaschke products. More precisely, for random finite Blaschke products with independent and identically distributed zeros with a common probability distributionμon the unit disc, the empirical measures of their critical points have been shown to converge weakly toμin probability. Consequently, this work contributes the very first probabilistic result to T. W. Ng and C. Y. Tsang's polynomial-finite-Blaschke-product dictionary. While the above works address probabilistic problems about zero ...