Tumbling mills and some theoretical and practical aspects of milling are discussed. Attention is directed towards the difficulty of relating the dynamical analyses of the motions of particles within mills to the existing treatments of breakage processes. This synthesis may be advanced by a study of milling that utilizes the matrix analysis of breakage processes. Application of the matrix analysis to results obtained on (i) grinding in open-circuit a zinc ore in a conical pebble mill and in a modified conical mill, (ii) "grinding of cement clinker in a compound mill, and (iii) to some drop-weight grindability tests is discussed. It is shown that the 'IT-breakage model adequately describes the breakage processes of each of the tests. (The proportion broken in each breakage cycle is given by 'IT and is independent of particle size.)The size distributions of the products are accurately predicted in the cases considered. Matrix analyses of the breakage of coals, various ores and a cement clinker have been made. Appendices give an elementary account of the matrix methods and give the equations fully describing reverse order closed circuit operations. The matrix analyses of breakage processes and the survey are used to examine briefly closed circuit grinding.INTRODUCTIONTumbling mills are slow-speed crushing and grinding machines which have been developed from some fairly simple types into many widely differing forms. Many of these versatile mills have proved extremely useful in ore dressing and mineral beneficiation, whereas others have been used in the cement industry and less extensively in the coal industry.The performance of even the simplest tumbling mill is very imperfectly understood, whilst the more difficult problems associated with the balancing of complex milling circuits and with the establishment of the optimum operating conditions for the whole of a complex circuit are even less well apprehended.Our knowledge of the performance of tumbling mills is rather sharply divided into two categories: the first consists...