Today's process modeling languages often force the analyst or modeler to straightjacket real-life processes into simplistic or incomplete models that fail to capture the essential features of the domain under study. Conventional business process models only describe the lifecycles of individual instances (cases) in isolation. Although process models may include data elements (cf. BPMN), explicit connections to real data models (e.g., an entity relationship model or a UML class model) are rarely made. Therefore, we propose a novel approach that extends data models with a behavioral perspective. Data models can easily deal with many-to-many and one-to-many relationships. This is exploited to create process models that can also model complex interactions between different types of instances. Classical multiple-instance problems are circumvented by using the data model for event correlation. The declarative nature of the proposed language makes it possible to model behavioral constraints over activities like cardinality constraints in data models. The resulting object-centric behavioral constraint (OCBC) model is able to describe processes involving interacting instances and complex data dependencies. In this paper, we introduce the OCBC model and notation, providing a number of examples that give a flavour of the approach. We then define a set-theoretic semantics exploiting cardinality constraints within and across time points. We finally formalize conformance checking in our setting, arguing that evaluating conformance against OCBC models requires diagnostics that go beyond what is provided by contemporary conformance checking approaches.