This paper considers the minimum mean p-th error (MMPE) estimation problem: estimating a random vector in the presence of additive white Gaussian noise (AWGN) in order to minimize an Lp norm of the estimation error. The MMPE generalizes the classical minimum mean square error (MMSE) estimation problem. This paper derives basic properties of the optimal MMPE estimator and MMPE functional. Optimal estimators are found for several inputs of interests, such as Gaussian and binary symbols. Under an appropriate p-th moment constraint, the Gaussian input is shown to be asymptotically the hardest to estimate for any p ≥ 1. By using a conditional version of the MMPE, the famous 'MMSE single-crossing point' bound is shown to hold for the MMPE too for all p ≥ 1, up to a multiplicative constant. Finally, the paper develops connections between the conditional differential entropy and the MMPE, which leads to a tighter version of the Ozarow-Wyner lower bound on the rate achieved by discrete inputs on AWGN channels.