  We construct bi-Lipschitz embeddings into Euclidean space for manifolds and
orbifolds of bounded diameter and curvature. The distortion and dimension of
such embeddings is bounded by diameter, curvature and dimension alone. Our
results also apply for bounded subsets of complete Riemannian manifolds, and
complete flat and elliptic orbifolds. Our approach is based on analysing the
structure of a bounded curvature manifold at various scales by specializing
methods from collapsing theory to a certain class of model spaces.
