A locally recoverable code (LRC code) is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert-Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and distance of LRC codes with multiple recovering sets.