In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave a strongly-explicit construction of infinite families of ($p+1$)regular Ramanujan Cayley graphs, for infinitely many primes p. In this paper we extend the work of LPS and its successors to bigraphs (biregular bipartite graphs): we investigate the combinatorial properties of various generalizations of the notion of Ramanujan graphs, define a notion of Cayley bigraphs, and give strongly-explicit constructions of infinite families of ($p^{3}+1, p+1$)-regular Ramanujan Cayley bigraphs, for infinitely many primes p. In addition, we present a pseudorandomness characterization of Ramanujan bigraphs, and a more general notion of biexpanders. We also show that the graphs we construct exhibit the cutoff phenomenon with bounded window size for the mixing time of non-backtracking random walks, and present some other applications, such as optimal unitary gates in quantum computation.