We have measured the 1f voltage noise in continuous metal films. At room temperature, samples of pure metals and bismuth (with a carrier density smaller by 105) of similar volume had comparable noise. The power spectrum SV(f) was proportional to V̄2Ωfγ, where V̄ is the mean voltage across the sample, Ω is the sample volume, and 1.0<~γ<~1.4. SV(f)V̄2 was reduced as the temperature was lowered. Manganin, with a temperature coefficient of resistance (β) close to zero, had no discernible noise. These results suggest that the noise arises from equilibrium temperature fluctuations modulating the resistance to give SV(f)~V̄2β2kBT2CV, where CV is the total heat capacity of the sample. The noise was spatially correlated over a length λ(f)~(Df)12, where D is the thermal diffusivity, implying that the fluctuations obey a diffusion equation. The usual theoretical treatment of spatially uncorrelated temperature fluctuations gives a spectrum that flattens at low frequencies in contradiction to the observed spectrum. However, the empirical inclusion of an explicit 1f region and appropriate normalization lead to SV(f)V̄2~β2kBT2CV[3+2ln(lw)]f, where l is the length and w is the width of the film, in excellent agreement with the measured noise. If the fluctuations are assumed to be spatially correlated, the diffusion equation can yield an extended 1f region in the power spectrum. We show that the temperature response of a sample to δ- and step-function power inputs has the same shape as the autocorrelation function for uncorrelated and correlated temperature fluctuations, respectively. The spectrum obtained from the cosine transform of the measured step-function response is in excellent agreement with the measured 1f voltage noise spectrum. Spatially correlated equilibrium temperature fluctuations are not the dominant source of 1f noise in semiconductors and discontinuous metal films. However, the agreement between the low-frequency spectrum of fluctuations in the mean-square Johnson-noise voltage and the resistance fluctuation spectrum measured in the presence of a current demonstrates that in these systems the 1f noise is also due to equilibrium resistance fluctuations.