TITLE: existence of a lifting
QUESTION [5 upvotes]: I was asking myself if given a covering projection $p: X' \rightarrow X$ and a continuous map $f:Y \rightarrow X$, does a lift $f: Y \rightarrow X'$ always exists? If no, could you please exhibit a counterexample?

REPLY [2 votes]: Generalizing the counterexample in the already existing answer, given any non-simply connected space $X$ and a loop $\gamma: S^1 \to X$ representing a nontrivial element of $\pi_1(X)$, $\gamma$ cannot be lifted to the universal cover $p : \tilde{X} \to X$ since otherwise it would be nullhomotopic in the cover, thus in the base. 
However, if $p : \tilde{X} \to X$ is a covering map, $f : Y \to X$ a map such that $f_*(\pi_1(Y)) \subset p_*(\pi_1(\tilde{X}))$, then $f$ can be lifted to $\tilde{X}$ along $p$. This is known as the map lifting lemma.