Let $X$ be a topological space. A universal covering space is a covering space $\tilde{X}$ of $X$ which is connected and simply connected.

If $X$ is based, with basepoint $x$ then a based cover of $X$ is cover of $X$ which is also a based space with a basepoint $x'$ such that the covering is a map of based spaces. Note that any cover can be made into a based cover by choosing a basepoint from the pre-images of $x$

The universal covering space has the following universal property: If $\pi:(\tilde X,x_0)\to(X,x)$ is a based universal cover, then for any connected based cover $\pi':(X',x')\to (X,x)$ there is a unique covering map $\pi'':(\tilde X,x_0)\to(X',x')$ such that $\pi=\pi'\circ\pi''$

Clearly, if a universal covering exists, it is unique up to unique isomorphism. But not every topological space has a universal cover. In fact $X$ has a universal cover if and only if it is semi-locally simply connected (for example, if it is a locally finite CW-complex or a manifold).