Evidence can be complex in various ways: e.g., it may exhibit structural complexity, containing information about causal, hierarchical or logical structure as well as empirical data, or it may exhibit combinatorial complexity, containing a complex combination of kinds of information. This paper examines evidential complexity from the point of view of Bayesian epistemology, asking: how should complex evidence impact on an agent's degrees of belief? The paper presents a high-level overview of an objective Bayesian answer: it presents the objective Bayesian norms concerning the relation between evidence and degrees of belief, and goes on to show how evidence of causal, hierarchical and logical structure lead to natural constraints on degrees of belief. The objective Bayesian network formalism is presented, and it is shown how this formalism can be used to handle both kinds of evidential complexity – structural complexity combinatorial complexity.