The lifting problem:

Suppose that $p: E\longrightarrow B$ is a surjective continuous map between topological spaces and $f: T \longrightarrow B$ is a given continuous map, a lift of $f$ is by definition a continuous map $\tilde{f}: T \longrightarrow E$ such that

$$p\circ \tilde{f} = f$$

The lifting problem involves giving sufficient conditions for the existence of the lift ${\tilde f}$. The main point here is of course the continuity of the lift. The significance of the problem can be understood from complex analysis.