Using the ku- and BP-theoretic versions of Astey’s cobordism obstruction for the existence of smooth Euclidean embeddings of stably almost complex manifolds, we prove that, for e greater than or equal to fi(n), the (2n + 1)-dimensional 2 e torsion lens space cannot be embedded in Euclidean space of dimension 4n i 2fi(n) + 1. (Here fi(n) denotes the number of ones in the dyadic expansion of a positive integer n.) A slightly restricted version of this fact holds for e < fi(n). We also give an inductive construction of Euclidean embeddings for 2 e -torsion lens spaces. Some of our best embeddings are within one dimension of being optimal. 1. Main results For a positive integer m let ”(m) and fi(m) denote, respectively, the exponent in the highest power of 2 dividing m, and the number of ones in the dyadic expansion of m. Let L 2m+1 (2 e ) stand for the (2m + 1)-dimensional 2 e -torsion lens space, the quotient of S 2m+1 by the standard (diagonal) action ofZ/2 e (viewed as a subgroup of the unit circle S 1 ). Unless explicitly noted otherwise, all (non)embedding results are to be understood in the smooth sense. Yet, except for a few low-dimensional cases (carefully pinpointed in the text), our results are within Haefliger’s metastable range 2m ‚ 3(n + 1) where, for a smooth closed manifold M n , the existence of a topological embedding M n ‰R m is equivalent to the existence of a smooth embedding M n ‰R m . We study the (Euclidean) embedding dimension of L 2m+1 (2 e ), that is, the dimension of the smallest Euclidean space where this manifold can be embedded. We get no new (non)embeddings for e = 1 (although we reconstruct some of the known optimal ones), as our methods are generalizations of ideas already used for real projective spaces. Instead, we seek to understand the role the exponent e plays in determining the Euclidean embedding dimension of L 2m+1 (2 e ).