Persistence probabilities concern how likely it is that a stochastic process has a long excursion above fixed level and of what are the relevant scenarios for this behavior. Power law decay is expected in many cases of physical significance and the issue is to determine its power exponent parameter. I will describe recent progress in this direction (jointly with Sumit Mukherjee), dealing with stationary Gaussian processes that arise from random algebraic polynomials of independent coefficients, as well as recent progress (jointly with Jian Ding and Fuchang Gao), on universality of the persistence power exponent for iterated partial sums (an exponent which was determined by Yakov Sinai, over 30 years ago).