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In this talk, I will introduce a new geometric inequality:Ìý the SphereÌýCovering Inequality. The inequalityÌý states thatÌýÌý the total areaÌý of twoÌý{\it distinct}Ìý surfaces with Gaussian curvature less than 1,ÌýwhichÌýareÌýalso conformal toÌý the Euclidean unit diskÌý with the same conformal factor on the boundary,Ìý must be at least $4 \pi$.Ìý In other words,Ìý the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang andÌýP. Yang.ÌýOther applications of this inequality include theÌýclassification of certain Onsager vorticesÌý on the sphere,Ìý the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori andÌý theÌýstandard sphere, etc.ÌýThe resolution of several open problems in theseÌýareas willÌý be presented.ÌýSome generalizations of the inequality toÌýinclude singular terms or more generalÌý surfaces will also be presented.