The moduli of vector bundles on a smooth curve is a beautiful illustration of many of the ideas and techniques that arise in algebraic moduli theory. In the past few years, a general theory of stability, $\Theta$-stratifications, and good moduli spaces for algebraic stacks has generalized many aspects of this moduli problem. However, questions of boundedness do not yet have a general answer -- this tends to be one of the trickiest parts of applying the machinery. I will discuss a new approach to boundedness for a vast generalization of the moduli of vector bundles on a curve: moduli spaces of objects in a dg-category that are semistable with respect to a Bridgeland stability condition. This involves reformulating the notion of a Bridgeland stability condition in a way that is especially convenient for moduli theory.