In the year 2000, Judea Pearl published Causality. What controversies surround this work? What are its major criticisms?
Some authors dislike Pearl's focus on the directed acyclic graph (DAG) as the way in which to view causality. Pearl essentially argues that any causal system can be considered as a non-parametric structural equation model (NPSEM), in which the value of each node is taken as a function of its parents and some individual error term; the error terms between different nodes may in general be correlated, to represent common causes.
Cartwright's book Hunting Causes and Using Them, for example, gives an example involving a car engine, which she claims cannot be modelled in the NPSEM framework. Pearl disputes this in his review of Cartwright's book.
Others caution that the use of DAGs can be misleading, in that the arrows lend an apparent authority to a chosen model as having causal implications, when this may not be the case at all. See Dawid's Beware of the DAG. For example, the three DAGs $A \rightarrow B \rightarrow C$, $A \leftarrow B \rightarrow C$ and $A \leftarrow B \leftarrow C$ all induce the same probabilistic model under Pearl's d-separation criterion, which is that A is independent of C given B. They are therefore indistinguishable based upon observational data.
However they have quite different causal interpretations, so if we wish to learn about the causal relationships here we would need more than simply observational data, whether that be the results of interventional experiments, prior information about the system, or something else.
I think this framework has a lot of trouble with general equilibrium effects or Stable Unit Treatment Value Assumption violations. In that case, the "untreated" observations no longer provide the desired counterfactual in a meaningful way. Massive job training programs that shift the entire wage distribution are one example. The counterfactual may not even be well-defined in some cases. In Morgan and Winship's Counterfactuals and Causal Models, they give an example of the claim that the 2000 election would have gone in favor of Al Gore if felons and ex-felons had been allowed to vote. They point out that the counterfactual world would have very different candidates and issues, so that you cannot characterize the alternative causal state. The ceteris paribus effect would not be the policy relevant parameter here.
The most important criticism of Pearl's system is, from my perspective, that it has not yielded any practical, empirical advances anywhere it has been used. Given how long it has been around, there's no reason to think it will ever be a practical tool. This indicates that it can be used for some theoretical and perhaps didactic purposes, but a practical researcher will gain little from studying it.