Perpetuity is usually defined as a random variable R satisfying the equation R=Q+MR, where the equality is meant to hold in distribution and where, on the right-hand side (Q,M) are random variables independent of R. Alternatively, R is a limit, in distribution, of a sequence (R_n) satisfying R_n=Q_n+M_nR_{n-1}, n > 0, where (Q_n,M_n) are iid copies of (Q,M), (Q_n,M_n) is independent of R_{n-1}, and R_0 is arbitrary. Conditions guaranteeing convergence in distribution of (R_n) have been given by Kesten in 1973. Random variables satisfying the above equation are ubiquitous in applied mathematics. The main focus of research has been on the tail behavior of R: P(|R| > x), as x goes to infinity. The case P(|M| > 1) > 0 was analysed by Kesten who showed that R is always heavy-tailed. The complementary case 0 < | M| < 1 is much less understood. Goldie and Grübel showed that in that case, the tails are never heavier than exponential and that if |M| behaves near 1 as a uniform random variable then the tails of R are Poissonian. In this talk we will present further results about the tails of R and their connection to the behavior of |M| near 1. This is a based on a joint work with Jacek Wesolowski, Technical University of Warsaw.