Operator learning has emerged as a powerful paradigm in scientific computing, with applications including surrogate modeling of partial differential equations and data-driven simulation of complex experimental systems, even in the absence of explicit governing equations. In this talk, I will discuss operator learning through the lens of statistical learning theory and identify several new learning-theoretic phenomena that arise in this setting. I will then focus on the role of data collection protocols and show that transitioning from passive (i.i.d sampling) to active data collection can fundamentally change which operator classes are learnable. Moreover, even for operator classes learnable under both protocols, active data collection can significantly improve sample efficiency, sometimes yielding exponential gains over passive approaches. I will conclude by highlighting open problems and exciting future directions in active data collection and, more broadly, in operator learning.