We study the complexity of computable and $\Sigma^0_1$ inductive definitions of sets of natural numbers. For example, we show how to assign natural indices to monotone $\Sigma^0_1$-definitions and then use these to calculate the complexity of the set of all indices of monotone $\Sigma^0_1$-definitions that are computable. We also examine the complexity of a new type of inductive definition, which we call weakly finitary monotone inductive definitions. Applications are given in proof theory and in logic programming.