In deconvolution in Rd; d 1; with mixing density p(2 P) and kernel h; the mixture density fp(2 Fp) can always be estimated with fn;n 2 P; via Minimum Distance Estimation approaches proposed herein, with calculation of fn's upper L1-error rate, an; in probability or in risk; h is either known or unknown, an decreases to zero with n: In applications, an is obtained when P consists either of products of d densities dened on a compact, or L1 separable densities in R with their dierences changing sign at most J times; J is either known or unknown. When h is known and p is ~q-smooth, vanishing outside a compact in Rd; plug-in upper bounds are then also provided for the L2-error rate ofn and its derivatives, respectively, in probability or in risk; ~q 2 R+; d 1: These L2-upper bounds depend on h's Fourier transform, ~h(6= 0); and have rates (log a??1 n )??N1 and aN2 n , respectively, for h super-smooth and smooth; N1 > 0; N2 > 0: For the typical an (log n) n??; the former (logarithmic) rate bound is optimal for any > 0 and the latter misses the optimal rate by the factor (log n) when = :5; > 0; > 0: The exponents N1 and N2 appear also in optimal rates and lower error and risk bounds in the deconvolution literature.