This paper providesan overview of ongoing studies in the area of thermocapillaryconvection driven by a surface tension gradient parallel to thefree surface in a floating zone, with emphases focused around theonset of oscillatory thermocapillary convection, also known as thetransition from quasisteady convection to oscillatory convection.The onset of oscillation depends on a set of critical parameters,and the margin relationship can be represented by a complexfunction of the critical parameters. The experimental resultsindicate that the velocity deviation of an oscillatory flow hasthe same order of magnitude as that of an average flow, and thedeviations of other quantities, such as temperature and freesurface radii fluctuations, are much smaller when compared withtheir normal counterparts. Therefore, the onset of oscillationshould be a result of the dynamic process in a fluid, and theproblem is a strongly nonlinear one. In the past few decades,several theoretical models have been introduced to tackle theproblem using analytical methods, linear instability analysismethods, energy instability methods, and unsteady 3D numericalmethods, the last of which is known to be the most suitable for athorough analysis of strong nonlinear processes, resultinggenerally in a better agreement with the experimental results. Thetransition from oscillatory thermocapillary convection toturbulence falls under the studies of chaotic behavior in a newsystem, which opens a fascinating new frontier in nonlinearscience, a hot research area drawing many recent works. Thispaper, with 93 references cited, reviews theoretical models andanalyses as well as experimental studies on thermocapillaryconnection in floating zones. DOI: 10.1115/1.2820798