Conditional logics play an important role in recent attempts to formulate theories of default reasoning. This paper investigates first-order conditional logic. We show that, as for first-order probabilistic logic, it is important not to confound statistical conditionals over the domain (such as "most birds fly"), and subjective conditionals over possible worlds (such as "I believe that Tweety is unlikely to fly"). We then address the issue of ascribing semantics to first-order conditional logic. As in the propositional case, there are many possible semantics. To study the problem in a coherent way, we use plausibility structures. These provide us with a general framework in which many of the standard approaches can be embedded. We show that while these standard approaches are all the same at the propositional level, they are significantly different in the context of a first-order language. Furthermore, we show that plausibilities provide the most natural extension of conditional logic to the first-order case: We provide a sound and complete axiomatization that contains only the KLM properties and standard axioms of first-order modal logic. We show that most of the other approaches have additional properties, which result in an inappropriate treatment of an infinitary version of the lottery paradox.