The continuous degrees measure the computability-theoretic content of elements of computable metric spaces, such as continuous real-valued functions on [0,1]. In 2004, I introduced the continuous degrees and proved that they properly extend the Turing degrees. Only later did I find out that Levin's "neutral measures", introduced in 1976, already give examples of non-Turing continuous degrees. Neutral measures are enigmatic; they are measures (say on Cantor space) for which every infinite binary sequence is random, even though tests have access to the measure itself. In this talk, I will discuss the continuous degrees and neutral measures. I will also explain connections to the topology of the Hilbert cube, the non-constructivity of Cantor's diagonal argument on the real numbers, and natural definability in the enumeration degrees.