What is the state of the art in capital theory?

Post-Keynesians still talk about it, but it’s mainly in reference to the historical controversy. (I haven’t followed it myself.)

I’ve seen some references to this paper:

Cohen, Avi J., and Geoffrey C. Harcourt. "Retrospectives: whatever happened to the Cambridge capital theory controversies?." Journal of Economic Perspectives 17.1 (2003): 199-214.

My impression as a non-economist is that there has not been much interest in the 21st century in these debates and it is not taught to students.

This is misconception but understandable one (I will get to the understandable part at the end). According to The New Palgrave Dictionary of Economics (Becker, 2017) - leading source for economic terminology - capital theory is defined thusly:

Capital theory examines the special role played by time in resource allocation studies. The determination of the interest rate and functional distribution of income as well as how rational agents invest are analysed within single- and multi-sector general equilibrium frameworks. Here, agents exercise perfect foresight over alternative consumption and capital accumulation programs. Efficient programs are characterized. Representative and multi-agent infinitely lived households are studied. Equivalence principles link the equilibrium programs and optimal paths. Heterogeneous agent models with borrowing constraints are reviewed. A behavioural model of intertemporal choice is also compared to its constant discounting counterpart.

Given the above broad definition of capital theory, virtually the whole corpus of modern macroeconomic theory is preoccupied with capital theory. The above broad definition of capital theory as a field would include models of economic growth such as the famous Solow-Swan model or Romer model and virtually any growth model that accounts for capital formation be it endogenous or exogenous. It would also include many overlapping generations models with dynamic capital formation and so on. For an overview of state of the art growth models you might want to look at Barro and Sala-i-Martin Economic Growth 2nd Edition (it is graduate level text so it has state of the art models), but really I can't think of a single macro - or even general economics textbook that would not at least briefly cover these issues (for example Blanchard et al. Macroeconomics: A European Perspective which is prominent macro textbook and deals with these issues).

In addition the above definition easily encompasses general equilibrium business cycle models as well. This is another large area of study in current macroeconomics. Presently this is theoretically covered mainly by the New Keynesian IS-LM models and Real Business Cycle theory. For state of the art examples of such models see Romer (2008) Advanced Macroeconomics 4th ed. For undergraduate treatment you can have look again at Blanchard et al. textbook mentioned earlier.

The areas mentioned above without any doubt the largest areas of research interest in macroeconomics (even throughout presently macroeconomics is becoming more empirical rather than relying on theoretical models as you can see in Angrist et al. 2017). Consequently, the research interest in capital theory can be presently argued to be quite strong and ubiquitous.

However, In my first paragraph I stated that the misconception you hold is understandable. This is because the term 'Capital Theory' fell out of favor and it is mostly used to describe particular historical sets of theories on capital (especially those you mentioned in your opening question) rather than an area of interest. Hence nowadays when someone publishes theoretical paper that will be about the theory of capital (e.g. Becker, and 1987 - just a random example) it will be often given different specific name rather than just capital theory (although occasionally I see people still using the term in non-historic context).

I am studying economics at university and we have to study a lot of various theories of capital in our courses. As muflon says this is taught quite a lot, but we were not taught much about the old debates just the new stuff. But what the other answer did not mention is that capital theory is also tough outside macroeconomics. For example in our inequality class we also had to study capital theory models.

On our syllabus Capital Theory and the Distribution of by C. J. Bliss was listed as further readings. I did not read the book myself but our professor said it is for those who want to study inequality in their PhD or do their masters thesis on inequality so I think it should cover state of the art models.