Resolving a question of Lévy, Wintner started the study of a Rademacher random multiplicative function in the 40's. This is a genuine multiplicative function supported on the squarefree integers such that its values at primes are given by ± 1 independent random variables. Several results concerning upper and omega bounds, low and high moments and central limit theorems have been proved. In this talk I will discuss sign changes of the partial sums of two models of random multiplicative functions: the Rademacher case and the completely multiplicative random case. I will also discuss the results in the literature about sign changes of the partial sums of the deterministic counterparts such as the Liouville and the Möbius function.