Many companies have adopted Process-aware Information Systems (PAIS) to support their business processes in some form. On the one hand these systems typically log events (e.g., in transaction logs or audit trails) related to the actual business process executions. On the other hand explicit process models describing how the business process should (or is expected to) be executed are frequently available. Together with the data recorded in the log, this situation raises the interesting question "Do the model and the log conform to each other?". Conformance checking, also referred to as conformance analysis, aims at the detection of inconsistencies between a process model and its corresponding execution log, and their quantification by the formation of metrics. This paper proposes an incremental approach to check the conformance of a process model and an event log. First of all, the fitness between the log and the model is measured (i.e., "Does the observed process comply with the control flow specified by the process model?"). Second, the appropriateness of the model can be analyzed with respect to the log (i.e., "Does the model describe the observed process in a suitable way?"). Appropriateness can be evaluated from both a structural and a behavioral perspective. To operationalize the ideas presented in this paper a Conformance Checker has been implemented within the ProM framework, and it has been evaluated using artificial and real-life event logs.