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The integrability (solvability via an associated single-valued linearproblem) of a differential equation is closely related to the singularitystructure of its solutions. In particular, there is strong evidence that allintegrable equations have the Painlev\'e property, that is, all solutions aresingle-valued around all movable singularities. In this expository article, wereview methods for analysing such singularity structure. In particular, wedescribe well known techniques of nonlinear regular-singular-type analysis,i.e. the Painlev\'e tests for ordinary and partial differential equations. Thenwe discuss methods of obtaining sufficiency conditions for the Painlev\'eproperty. Recently, extensions of \textit{irregular} singularity analysis tononlinear equations have been achieved. Also, new asymptotic limits ofdifferential equations preserving the Painlev\'e property have been found. Wediscuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer  School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.  Grammaticos and K. Tamizhman

Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property