The concept of k-regular maps was introduced into topology by Karol Borsuk in 1950′. Long before it was investigeted in interpolation and approximation theory by (among others) Chebyshev and Kolmogorov, and had distinctly applied mathematics flavor.

A continuous map f: X\to V, from a topological space to a vector space is k-regular of for any k distinct points in X their images are linearly independent.

Example: 
If f:X\to V is an embedding, then the map F: X\to R\oplus V, defned by F(x)=(1,f(x))
is 2-regular.

In analogy to theory of embeddings, one of central problems in the study of k-regular maps is (given k, X)  to construct a k-regular map of X into space V of minimal possible dimension. Unlike for embeddings, this is interesting already for X=R^d I will discuss  lower bounds (obstructions) to the existence to the existence of k-regular maps (this involves some algebraic topology of configuration spaces, following work of many people, most recent being Blagojevic, Cohen, Lueck and Ziegler) and upper bounds (constructions) (this involves some algebraic geometry of secant varieties and  Hilbert schemes, following the work of Buczynski, Januszkiewicz, Jelisiejew and Michalek).