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We examine the validity of the principle of mass conservation for solutionsof some typical equations in the theory of nonlinear diffusion, includingequations in standard differential form and also their fractional counterparts.We use as main examples the heat equation, the porous medium equation and the$p$-Laplacian equation. Though these equations have the form of conservationlaws, it happens that in some ranges, and posed in the whole Euclidean space,the solutions actually lose mass in time, they even disappear in finite time.This is a surprising fact.  In Part 1 we pay attention to the connection between the validity of massconservation and the existence of finite-mass self-similar solutions, as wellas the role in the asymptotic behaviour of more general classes of solutions.We examine the situations when mass conservation is replaced by its extremealternative, extinction in finite time.  The conservation laws offer difficult borderline cases in the presence ofcritical parameters that we identify. New results are proved. We establish massconservation in those pending cases. We also explain the disappearance of thefundamental solutions in a very graphical way.  The sections of Part 2 are devoted to the discussion of mass conservation forsome fractional nonlinear diffusion equations, where the situation is surveyedand a number of pending theorems are proved. We conclude with a long review ofrelated equations and topics in Part 3.  Summing up, the paper aims at surveying an important topic in nonlineardiffusion; at the time we solve a number of open problems on key issues andpoint out new directions.

A survey on mass conservation and related topics in nonlinear diffusion