The evolution of the universe is described by the physics of general relativity, which was discovered by Albert Einstein in the early 20th century. When compared to Newtonian physics, this theory provides a radically different framework for the physical description of the gravitational force.
In the Newtonian interpretation (where celestial bodies move according to the laws of Newton), space and time are absolute, with time no more than a parameter in the equations of motion. Meanwhile, gravity plays the role of a mysterious force of attraction between massive bodies.
The physics of general relativity is conceptually distinct--even if its equations of motion can be reduced to Newtonian equations in many practical cases, such as with respect to the motion of the moon, or, as we will see shortly, the overall evolution of the universe.
In general relativity, space and time are merged into one four-dimensional grid, whose properties are uniquely specified (via gravity) by the bodies inhabiting them. Gravity curves the spacetime grid, so general relativity thus describes gravitational interactions as manifestations of the spacetime curvature. Objects "fall under gravity" from less curved parts of spacetime to more curved parts of the spacetime. (When spacetime becomes infinitely curved, as in the case of black holes, the gravitational force is so strong that spacetime closes on itself, creating what is called a singularity in the fabric of the underlying spacetime continuum. Nothing can escape such objects.)
According to Einstein's general relativity equations, the spacetime containing matter cannot remain stationary and must either expand or contract. Galaxies and other sources, then, are not strictly expanding away from each other but rather are attached to the fixed grid on the expanding fabric of spacetime. Thus, the galaxies give us the impression of moving away from each other. Imagine the surface of a balloon, on which you put dots. Then start inflating the balloon. The distances between the dots will increase, so if you live in one of these dots, you will interpret this as the dots--which represent galaxies in this example--moving away from each other. In reality, of course, they remain in the same positions, with respect to latitudes and longitudes on the balloon, and it is the fabric of the balloon that is actually expanding.
In Newtonian physics, one can construct a mathematical analogy to the expansion of the universe by defining a system that is expanding or contracting under its own gravity, such as a galaxy made of stars or the solar system. In this framework, however, this expansion is not linked to stretching the fabric of any spacetime. Instead, space is some abstract absolute and fixed entity that all objects move through without affecting it. Thus one can ask not only "Where is the universe expanding to?" in the Newtonian framework, but also "What happened before the initial push?"
In the framework of general relativity, however, both of these questions become meaningless. Asking the question, "Where is the universe expanding to?" implies some other coordinate grid outside spacetime. But since spacetime is linked to matter, there is no outside to the surface of the balloon. Rather, it is all the spacetime that is available.