When is covariance a reasonable measure of correlation? I am brushing up on statistics for a data science course.
I saw an example of variables that are highly dependent, but still have 0 covariance (http://en.wikipedia.org/wiki/Covariance#Uncorrelatedness_and_independence)
Now ... what I want to know is: when would covariance be a good measure of correlation, and when not. What kind of data is suitable for analysis using covariance, and what isn't? 
 A: Clearly correlation, meaning here precisely Pearson correlation as a measure of closeness to linear relationship, is the covariance of two variables divided by the product of the corresponding standard deviations. So, there is one obvious answer:
1) Correlation and covariance coincide numerically if and only if the product of standard deviations is 1 in the same units of measurement, which is unlikely but not impossible. This is a mathematical property without much statistical application, except occasionally in simplifying calculations in theoretical examples. 
A more helpful property is that 
2) Covariance being positive, zero or negative implies the same  property for correlation. If there were interest only in signs of (linear) relationship, then looking at the sign of covariance would be informative. 
On the whole, my short answer is 
3) Rarely. Much of the point of correlation is that we need to standardise by division of the product of standard deviations to be able to assess strength of relationship. Even in #2 above knowing that a covariance is say 42,000 mm year is not very informative without dividing by the corresponding product of standard deviations with the same units. 
Your question implies that you may want to interpret "correlation" as "dependence" generally, which is ill-advised here. But broadly speaking, if Pearson correlation often fails to echo nonlinear dependence between two variables, then covariance can do no better and will usually be worse. 
