Let X be a closed subset of a manifold without boundary Q. A codimension 0 submanifold M of Q is a mapping cylinder neighborhood of X if it is a closed neighborhood of X in Q and if there is a proper map r: AM-P X such that M is homeomorphic to the mapping cylinder of r, fixing AM and X in the natural way. For example, regular neighborhoods of locally finite complexes in PL manifolds are mapping cylinder neighborhoods. We do not insist that M be collared in Q since this can always be arranged by an isotopy of M into itself along the structure lines. Edwards ([5]) contains a lot of information about mapping cylinder neighborhoods. It follows immediately from the definition that subsets possessing mapping cylinder neighborhoods are retracts of the neighborhoods, hence are finite dimensional ANR's. The converse question, whether finite dimensional ANR's have mapping cylinder neighborhoods, at least for some embeddings in manifolds, is interesting. If they do, then compact, finite dimensional ANR's have finite homotopy type, since these neighborhoods are compact manifolds, and so have finite homotopy type by Kirby-Siebenmann (17]). We prove a partial result in the direction of the converse, namely, if X is a locally compact ANR, embedded as a closed, 1-LC codimension 4 subset of a finite dimensional manifold without boundary Q, then X x M, has a mapping cylinder neighborhood in Q x M, where either M. M= SI, or (M, Mj) = (R', [0, ao)). Our proof also works in infinite dimensions and yields the fact that if X is a locally compact ANR which is embedded as a closed Z-set in a Hilbert cube manifold Q, then X x M. has a cylinder neighborhood in Q x M. Taking suitable compactifications in the case (M, Mj) = (Ri, [0, ao)), one finds that the cone on X is the image of the cone on Q by a map with contractible point inverses. J. West ([10]) has taken this last form of our result and shown that in the infinite dimensional setting it implies X actually has a mapping cylinder