We propose a new technique to estimate the signal-to-noise ratio (SNR) over the flat-fading channel. This is a non-data-aided, envelope-based estimator that can be applied to non-constant modulus constellations which is a feature not found in existing approaches. We also analyze the performance of our estimators for both PSK and non-PSK constellations by deriving their asymptotic variances and comparing with the Cramer-Rao Bounds (CRBs). Moreover, we discuss how the SNR estimates can be used to approximate the bit error rate (BER) and how the accuracy of the SNR estimate is related to that of the BER estimate, which justifies the necessity for the accurate estimation of SNR. The analytical performance is shown for both PSK and non-PSK constellations, such as 8QAM, and 16QAM. Monte Carlo simulation results corroborate our analysis.