In　2018,　a　UCLA　research　group　published　an　important　paper　on　optical　neural　network　(ONN)　research　in　the　journal　Science.　It　developed　the　world＇s　first　all-optical　diffraction　deep　neural　network　(DNN)　system,　which　can　perform　MNIST　dataset　classification　tasks　at　near-light-speed.　To　be　specific,　the　UCLA　research　group　adopted　a　terahertz　light　source　as　the　input,　established　the　all-optical　diffractive　DNN　(D2NN)　model　using　the　Rayleigh-Sommerfeld　diffraction　theory,　optimized　the　model　parameters　using　the　stochastic　gradient　descent　algorithm,　and　then　used　3D　printing　technology　to　make　the　diffraction　grating　and　built　the　D2NN　system.　This　research　opened　a　new　ONN　research　direction.　Here,　we　first　review　and　analyze　the　development　history　and　basic　theory　of　artificial　neural　networks　(ANNs)　and　ONNs.　Second,　we　elaborate　D2NN　as　holographic　optical　elements　(HOEs)　interconnected　by　free　space　light　and　describe　the　theory　of　D2NN.　Then　we　cover　the　nonlinear　research　and　application　scenarios　for　D2NN.　Finally,　the　future　directions　and　challenges　of　D2NN　are　briefly　discussed.　Hopefully,　our　work　can　provide　support　and　help　to　researchers　who　study　the　theory　and　application　of　D2NN　in　the　future.　©　2023　Optica　Publishing　Group.