We present the results of the analysis of neutrino observations by the Antarctic Muon and Neutrino Detector Array(AMANDA) correlated with photon observations of more than 400 gamma-ray bursts (GRBs) in the northernhemisphere from 1997 to 2003. During this time period,AMANDA’s effective collection area for muon neutrinos waslarger than that of any other existing detector. After the application of various selection criteria to our data, we expect1 neutrino event and<2 background events. Based on our observations of zero events during and immediately priorto the GRBs in the data set, we set the most stringent upper limit on muon neutrino emission correlated with GRBs.Assuming a Waxman-Bahcall spectrum and incorporating all systematic uncertainties, our flux upper limit has anormalization at 1 PeVof E2 6:3 ; 109 GeV cm2 s1 sr1, with 90% of the events expected within the energyrange of 10 TeV to 3 PeV. The impact of this limit on several theoretical models of GRBs is discussed, as well asthe future potential for detection of GRBs by next-generation neutrino telescopes. Finally, we briefly describe severalmodifications to this analysis in order to apply it to other types of transient point sources.