Are medical screening tests valuable?

Are medical screening tests valuable?  The answer isn't always as clear as you might think.  Study the chart for breast cancer screening (mammograms) below to see one example of this (more details here).  Some kinds of screening (example: colonoscopy) themselves pose a risk, and though that risk is generally quite small so are the positive detections.  Mammograms involve exposure to high-energy radiation, which is a known risk for causing cancer.  All kinds of screening have a false-positive problem – and these false positives put the patient through a lot of stress and additional testing, which also has risks. 

This stuff is hard.  Some kinds of medical screening absolutely make sense.  Some of them absolutely do not make sense.  Most of them are a lot less clear...