We summarize the current status of cosmological measurements using SNe Ia .
Searches to an average depth of z \sim 0.5 have found approximate 100 SNe Ia to date , and measurements of their light curves and peak magnitudes find these objects to be about 0 ^ { m } .25 fainter than predictions for an empty universe .
These measures imply low values for \Omega _ { M } ( 0.2-0.3 ) and a positive cosmological constant , with high statistical significance .
Searches out to z \sim 1-1.2 for SNe Ia ( peak magnitudes m _ { I } \sim 24.5 ) will greatly aid in confirming this result , or demonstrate the existence of systematic errors .
Multi-epoch spectra of SNe Ia at z=0.5 are needed to constrain possible evolutionary effects .
I band searches should be able to find SNe Ia out to z \sim 2 .
We discuss some simulations of deep searches and present histograms of type Ia and type II SNe discovery statistics at several redshifts .