Several methods have been developed to assess the thermal state of the mantle below oceanic ridges, islands, and plateaus, on the basis of the petrology and geochemistry of erupted lavas. One leads to the conclusion that mantle potential temperature (i.e., TP) of ambient mantle below oceanic ridges is 14300C, the same as Hawaii. Another has ridges with a large range in ambient mantle potential temperature (i.e., TP = 1300--15700C), comparable in some cases to hot spots (Klein and Langmuir, 1987; Langmuir et al., 1992). A third has uniformly low temperatures for ambient mantle below ridges, ~13000C, with localized 2500C anomalies associated with mantle plumes. All methods involve assumptions and uncertainties that we critically evaluate. A new evaluation is made of parental magma compositions that would crystallize olivines with the maximum forsterite contents observed in lava flows. These are generally in good agreement with primary magma compositions calculated using the mass balance method of Herzberg and O'Hara (2002), and differences reflect the well-known effects of fractional crystallization. Results of primary magma compositions we obtain for mid-ocean ridge basalts and various oceanic islands and plateaus generally favor the third type of model but with ambient mantle potential temperatures in the range 1280--14000C and thermal anomalies that can be 200--3000C above this background. Our results are consistent with the plume model.