When a client consumes a cloud service, computational liabilities are transferred to the service provider in accordance to the cloud paradigm, and the client loses some control over software components. One way to raise assurance about correctness and dependability of a consumed service and its software components is monitoring. In particular, a monitor is a system that observes the behavior of another system, and observation points that expose the target system's state and state changes are required. Due to the cloud paradigm, popular techniques for monitoring such as code instrumentation are often not available to the client because of limited visibility, lack of control, and black-box software components. Based on a literature review, we identify potential observation points in today's cloud services. Furthermore, we investigate two cloud-specific monitoring applications based on our ongoing research. While service level agreement (SLA) monitoring ensures that agreed-upon conditions between clients and providers are met, language-based anomaly detection monitors the interaction between client and cloud for misuse attempts.