How many times does one need to shuffle a deck of cards to randomize them? This question has been throughly studied for many types of card shuffling, with connections to Markov chains, representation theory, and group theory. In this talk, we will discuss a weaker version of this topic. What if we only need certain “features” of the deck to be close to uniformly random? This is typically considered through the lens of permutation statistics. We will mention some corresponding results on riffle shuffles by many authors. Then we will give an overview of some of our recent work in this area on random-to-top (and related) shuffling.