The classic algorithm of Viterbi computes the most likely path in a Hidden Markov Model (HMM) that results in a given sequence of observations. It runs in time O(Tn^2) given a sequence of T observations from a HMM with n states. Despite significant interest in the problem and prolonged effort by different communities, no known algorithm achieves more than a polylogarithmic speedup. In this paper, we explain this difficulty by providing matching conditional lower bounds. Our lower bounds are based on assumptions that the best known algorithms for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique problem in edge-weighted graphs are essentially tight. Finally, using a recent algorithm by Green Larsen and Williams for online Boolean matrix-vector multiplication, we get a 2^{Omega(sqrt{log n})} speedup for the Viterbi algorithm when there are few distinct transition probabilities in the HMM.