Finding new ways of extracting and analyzing useful information from exploding volumes of unstructured and semi-structured text sources has become one of the greatest challenges in the era of big data. After new technologies have enabled efficient solutions for collecting and storing these data, the next step in computer science research is to develop scalable approaches for efficient analysis of dynamics in text streams. This dissertation addresses this challenge by examining how visual analytics can help the users gain new insights from systems for explorative analysis of events in text streams that are more efficient and easier to use. My work revolves around the concept of streaming visual analytics, whose goal is to combine resource constraints of the computer and time constraints of the user to provide more scalable tools. I identify challenges in the user, data, and visualization domain, discuss open issues, and derive design considerations to help practitioners in developing future systems for incremental data. Based on this approach, I describe novel visual analytics methods for detection and exploration of events in news streams: CloudLines, a compact overview visualization for events in multiple event sequences in limited space, and Story Tracker, a visual analytics system for exploration of news stories development and their complex properties. Novel experimental visualizations are introduced to demonstrate the applicability of the approach in real-time monitoring scenarios. I describe how the streaming visualization concepts pervade my work and outline directions for future research.