Where is the center of the universe?
According to all current observations, there is no center to the universe. For a center point to exist, that point would have to somehow be special with respect to the universe as a whole. Let us think about all the different types of effects that could create a center.
First, if an object is rotating, you can define a center of rotation. The center of rotation is the one spot on a rotating object that is stationary. For the earth, the center of rotation is the axis connecting the North and South pole. For a basketball player spinning a basketball on his finger, the center of rotation is the point where the ball touches his finger. The center of rotation for a wheel on an axle is the center of the axle. Observations of the universe have not found any rotation at all to the universe as a whole. With no rotation, there is no center of rotation.
Next, you can define a center of mass. If an object is finite, the center of mass is just the point that, on average, has an equal amount of mass surrounding it in all directions. The situation gets more complicated for an infinite object. If an object is infinite and uniform, you simply cannot define a center of mass because all points are identical. On the other hand, if an object is infinite but not uniform (for instance it has a single knot of high density at one point), you can define the center of mass of the entire object as the center of mass of the non-uniformity. For instance, consider a cloud in the sky. Certain kinds of clouds don’t have a well-defined boundary, but instead just stretch out in all directions, getting thinner and thinner. Even though the cloud stretches out effectively to infinity, the high density region of the cloud exists in a limited volume, so you can find a center of mass through a limiting procedure. Observations currently indicate that the universe is infinite in size. Although planets and stars do represent non-uniformities in the spacetime structure, on the universal scale, such uniformities are randomly dispersed. On average, therefore, the universe is uniform. Being infinite and uniform, there is no way to define a center of mass for the universe.
Another possibility is a center of charge. Similar to the center of mass, this would be a point in an object where the amount of electric charge is on average the same in all directions surrounding it. The center of charge for a uniformly charged sphere would just be the center of the sphere. Similar to the mass distribution, the charge distribution of the universe is infinite and uniform on average so that there is no center of charge.
Next, there could be a center of curvature. Like a salad bowl, there could be a central point to the universe from which all other points curve away from. But current observations have found the universe to be flat and not curved at all.
Yet another possibility is a center of expansion. If you bolt a rubber sheet to the ground and then have people pull on all sides, the place where the sheet is bolted becomes the center of expansion. The center of expansion is the point in space from which all other points are moving away. A wealth of astronomical observations has revealed that the universe is indeed expanding. These observations are the foundation for the concept that a Big Bang started the universe. Because the universe is expanding, if you run time backwards, there had to be a time when the universe was all compacted to one point. Since the universe is expanding, you would think there is a center of expansion. But observations have revealed this not to be the case. The universe is expanding equally in all directions. All points in space are getting uniformly distant from all other points at the same time. This may be hard to visualize, but the key concept is that objects in the universe aren’t really flying away from each other on the universal scale. Instead, the objects are relativity fixed in space, and space itself is expanding. You might be tempted to say that the location of the Big Bang is the center of the universe. But because space itself was created by the Big Bang, the location of the Big Bang was everywhere in the universe and not at a single point. The major aftereffect of the Big Bang was a flash of light known as the Cosmic Background Radiation. If the Big Bang happened at one location in space, we would only see this flash of light coming from one spot in the sky (we can see a flash that happened so long ago because light takes time to travel through space and the universal scale is so big). Instead, we see the flash as coming equally from all points in space. Furthermore, once the motion of the earth is accounted for, the flash of light is equally strong in all directions on average. This indicates that there is no center of expansion.
Another way to define a center would be to identify some object or feature that exists only at one spot, such as a supermassive black hole or super-large nebula. But observations indicate that all types of objects are randomly peppered through the universe.
No matter how we try to define and identify it, the universe simply has no center. The universe is infinite and non-rotating. Averaged over the universal scale, the universe is uniform.