The question of whether stable minimal surfaces are holomorphic under suitable geometric conditions has been much studied, beginning with work of Lawson-Simons in complex projective space, and the proof of the Frankel conjecture by Siu-Yau. A theorem of Micallef shows that a stable minimal immersion of a complete oriented parabolic surface into Euclidean 4-space is holomorphic with respect to an orthogonal complex structure, and the same result in all dimensions if additionally the minimal surface has finite total curvature and genus zero. However, Arezzo, Micallef and Pirola gave a counterexample in general. It is therefore necessary to strengthen the stability condition in the general question. We will discuss some recent progress on this question. This is joint work with R. Schoen.