If I am not mistaken, no explicit metric on a compact Calabi-Yau manifold is known. I guess part of the difficulty is due to the fact that compact Calabi-Yau manifolds do not admit continuous isometries and therefore there is no adapted system of coordinates that can be used to write down the metric in a simpler, explicit way. In fact, in the non-compact case there are explicit examples of Calabi-Yau metrics.
Now, compact Calabi-Yau manifolds are extremely important in String/M/F-theory: they are for example the compactification spaces of fluxless String-Theory compactifications and they are the general compactification spaces of F-theory compactifications, which are very relevant from the phenomenological point of view. Knowing an explicit Calabi-Yau metric would be interesting, because among many other things then one could explicitly know the non-linear sigma model of the string propagating in that background.
My question is then: is this actually an active field of research? What is the actual status of the problem? I don't recall seeing many papers (or any) oriented towards solving this problem and finding an explicit metric.