What are the empirical techniques to show causation? A simple linear regression only shows correlation between two variables. To establish causation, two commonly taught methods are IV regression and natural experiments. What are the other methods people use to establish causation?
Natural experiments are usually a setting for causal inference rather than a causal inference tool per se. You often need to employ something like difference-in-difference or instrumental variables anyway even when you have a natural experiment.

 Here a list of statistical causal inference approaches (Approach: Lay description) 


Instrumental Variables: Randomly assigned variable X influences Z only through Y 
Difference in Differences:  If two groups have a common trend and only one group is treated then the change in the difference between the groups is the treatment effect
Regression Discontinuity: If a hard threshold determined treatment, look at difference right around that threshold
Propensity score matching: Create a control group by matching untreated observations that were likely to be treated (but not in fact treated) with treated observations with a similar probability of treatment. 
Manhalobis distance matching:   Create a control group by matching untreated observations that look similar to treated. Another notable distance measure is Coarsened Exact Matching.  
Synthetic control: When you have only one treated observation, create a composite of untreated observations that individually are imperfect controls but collectively act as a good control.
Synthetic Cohorts - Treat repeated observations of groups like a panel of individuals and employ panel techniques. 
Selection bias modeling like Heckman correction: Assume a parametric form for selection bias and remove it, so the corrected regression results can be interpreted causally.
Sample weighting more broadly - fix bias resulting from endogenous participation and un-modeled heterogeneity by weighting sample units to look more like the true population of interest.