We describe the application of high accuracy Srspectroscopy to the measurement of the variation of thefundamental constants of nature. We first describe recent progressof the JILA Sr optical frequency standard, with a systematicuncertainty evaluation at the 10-16 fractional frequencylevel. Using recent internationally based measurements of the Srclock frequency, we show improved constraints of gravitational andtemporal changes in the fine structure constant and theelectron-proton mass ratio. Finally, we describe how ultracoldatomic strontium, confined in an optical lattice, can beassociated into molecular dimers and be used for amodel-independent measurement of the variation of theelectron-proton mass ratio.