Is simultaneity well-defined in general relativity? In special relativity for each event and reference frame we can find a plane of simultaneous events.
I wonder is it possible to do the same in general case in curved space? Is simultaneity even meaningful in GR?
 A: Say we have a congruence of observers with some 4-velocity field $u^{\alpha}$. Locally each observer has a simultaneity plane determined by the Einstein synchronizaton convention just as in SR. Now if these observers can Einstein synchronize their clocks with their infinitesimal neighbors we can "patch" these local simultaneity planes together to make a one-parameter family of global simultaneity surfaces with a global time coordinate that all the observers in the congruence agree on. 
This is, as far as I know, the closest one can get to the usual notion of simultaneity in inertial frames in SR. Keep in mind this isn't really special to GR; one can encounter subtleties regarding simultaneity even in rotating frames in SR for example.
Anyways, the condition for such a foliation to exist is the usual Frobenius condition that $u_{[\alpha}\nabla_{\beta}u_{\gamma]} = 0$.   
A: I believe there are a few candidates notions, but it's not obvious which is the right one. 
You can talk about surfaces that are everywhere orthogonal to a timelike Killing field. 
If there isn't a timelike Killing field you can always talk about the surface geodesically generated by a timelike vector at $p$: the set of points on geodesics passing through $p$ that are orthogonal to that vector.
I believe both definitions give you planes of simultaneity in a flat Minkowski spacetime, but they can come apart in the more general setting.
(People sometimes talk about Cauchy surfaces -- surfaces that intersect every timelike or null curve exactly once. But these are not equivalent to planes of simultaneity in Minkowski space.)
A: Simultaneity has absolutely no meaning whatsoever in GR. In SR, we say simultaneity is relative and can't be trusted. In GR, we don't even say that. You might occasionally catch someone using the word "simultaneous", but that doesn't mean what you think. In GR, you can take any event, any reference frame, any set of values and define some arbitrary time-like coordinate that foliates spacetime such that almost anything is "simultaneous". Then, you can change the gauge and the whole notion of time shifts.
What we normally do is choose a metric with some coordinate time. Then we can "say" that all events at the same coordinate time are a "hypersurface of constant time", which is the GR equivalent of "simultaneous". However, you could choose some other time-dependent value. For instance, in inflationary cosmology, we might have a metric with some coordinate time, $t$, and an inflaton field, $\phi$. We might evolve through inflation with respect to constant $t$ hypersurfaces, but then we call the time when $\phi$ reaches some value corresponding to the end of inflation to be simultaneous, even if it isn't a constant $t$ hypersurface. We do this because we want to say that inflation ends everywhere at the same time, which means we are now defining a constant $\phi$ hypersurface as simultaneous. This isn't even a difference in reference frames, all we have to do is choose a different gauge for our metric and we can define a new concept of simultaneity.
So to answer your question, no, simultaneity is a meaningless idea in curved space. What we use is constant hypersurfaces; we choose a parameter that has a desired foliation of spacetime and then slice spacetime such that every point in that slice has the same value of the chosen parameter.
You can arbitrarily define a coordinate time and take its constant hypersurfaces as simultaneous, but that's even less meaningful than in SR.
A: "Simultaneity" has a well-defined technical meaning in SR, as well as in GR.
What I am referring to is the "Einstein’s convention for the definition of simultaneity", namely the set of events that are considered by an observer $O$ as being simultaneous to a given reference event on its worldline. The answer of @FenderLesPaul says this in a more refined mathematical way.
Simultaneity in GR is a generalization of the same concept in SR: given an observer $O$, it is possible to define a 3D space like sub-manifold called "present of $O$"(this is the manifold that is orthogonal to the 4-velocity of the observer $O$). Events in the present of $O$ are "simultaneous" (not simultaneous in an absolute way, but simultaneous according to $O$).
In GR the "present" of a single observer is defined "in its vicinity": since the spacetime of GR is locally flat, the same construction of SR is applied in the so-called "tangent space". If you want to extend the concept everywhere (not only in the vicinity of $O$), then you need a whole family of observers (namely, a family of world lines that fulfil the Frobenius condition).
To dig more, I suggest you to have a look at "3.2 Observers, Lorentz Factors and Relative Velocities" here: https://arxiv.org/abs/gr-qc/0603009
