We have constructed daily global long- and short-term earthquake forecasts for CSEP (Collaboratory for the Study of Earthquake Predictability). We forecast the earthquake rate per the unit area, time, and magnitude on a 0.5 degree grid for a global region between 75N and 75S latitude (301 by 720 grid cells). We use two global earthquake catalogs: the CMT catalog and the PDE catalog. Similar to our previous forecasts, the new forecasts are based on the smoothed maps of past seismicity and assume spatial and temporal clustering. The forecast based on the CMT catalog (with the magnitude threshold 5.8) includes an estimate of a focal mechanism of future earthquakes and of the mechanism uncertainty. The forecasted tensor focal mechanism makes it possible to calculate an ensemble of seismograms for each point of interest on the Earth's surface. The application of the present methodology to other subduction zones does not present any fundamental difficulty, and these subduction events constitute a major part of the global seismicity. We also introduce a new approach that circumvents the need for focal mechanisms. This permits the use of the PDE catalog that reliably reports many smaller quakes with a higher location accuracy. The result is forecasting earthquakes at a higher spatial resolution and down to a magnitude threshold of 5.0. Such new forecasts can be tested within a relatively short period of time since smaller events occur with greater frequency. Efficiency of the forecasts can be measured by their probability gains per earthquake compared to spatially or temporally uniform Poisson distribution of events. For the short-term forecast the gain is about 2.0 for the CMT catalog and 3.7 for the PDE catalog. Preliminary tests indicate that for the long-term forecast the gain is of the order 20-25.