Bernoulli convolutions are distributions on the real line obtained as power series $\sum_{n>0} \pm \lambda^n$ with random signs. They are basic examples of self-similar measures. Determining for which value of the parameter $\lambda$ is the resulting measure absolutely continuous is an open problem with a long history going back to Erd\H{o}s. New methods, blending entropy theory and Diophantine analysis have been used in recent years to tackle it and make new advances to neighboring problems such as the study of random polynomials of large degree. The lecture will give an overview of these developments.