Abstract: Many mathematical modelshavebeen developed sofar, butthereisstill much tobe donefor mathematical modelingin thefield of initialsolidification. This paper reviewsthe mecha nism of irregular solidification and mathematical modeling of irregular solidification andlongitudinalcrackformation. Main responsiblefactorsforirregularsolidification areconsid eredtobe nucleationofsolid phaseonthe moldsurfaceandshelldeflection duetothermalcon traction ofshell. Someofrecent worksonshellirregularity andcrackformation arepresented including a new model of shell irregularity, in which heat flow and shell deflection are linked in the calculation, and a strain model which takes account of frictional forcein the transverse direction. In orderto develop a quantitative modelto predictlongitudinalcrack, fartherstudy willbeexpected in revising and linkingthese models.