Statistical inference is at the heart of the scientific method, and the likelihood function is at the heart of statistical inference. However, many scientific theories are formulated as mechanistic models that do not admit a tractable likelihood. While traditional approaches to confronting this problem may seem somewhat naive, they reveal numerous other considerations in the scientific workflow beyond the approximation error of the likelihood. I will highlight how normalizing flows and other techniques from machine learning are impacting scientific practice, discuss current challenges for state-of-the-art methods, and identify promising new directions in this line of research.