We consider the phase retrieval problem of reconstructing a -dimensional real or complex signal from (possibly noisy) observations , for a large class of correlated real and complex random sensing matrices , in a high-dimensional setting where while . First, we derive sharp asymptotics for the lowest possible estimation error achievable statistically and we unveil the existence of sharp phase transitions for the weak-and full-recovery thresholds as a function of the singular values of the matrix . This is achieved by providing a rigorous proof of a result first obtained by the replica method from statistical mechanics. In particular, the information-theoretic transition to perfect recovery for full-rank matrices appears at (real case) and (complex case). Secondly, we analyze the performance of the best-known polynomial time algorithm for this problem---approximate message-passing---establishing the existence of statistical-to-algorithmic gap depending, again, on the spectral properties of . Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval for a broad class of random matrices.