A central theme in distributed network algorithms concerns understanding and coping with the issue of {\em locality}. Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for \emph{distributed decision problems}. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard model of computation and define (for {\em local decision}) as the class of decision problems that can be solved in communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class , containing all languages for which there exists a randomized algorithm that runs in rounds, accepts correct instances with probability at least and rejects incorrect ones with probability at least . We show that is a threshold for the containment of in . More precisely, we show that there exists a language that does not belong to for any but does belong to for any such that . On the other hand, we show that, restricted to hereditary languages, , for any function and any such that . In addition, we investigate the impact of non-determinism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that non-determinism does help, but that this help is limited, as there exist languages that cannot be decided non-deterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with non-determinism that enables to decide \emph{all} languages \emph{in constant time}. Finally, we introduce the notion of local reduction, and establish some completeness results.