This is joint work with Peter Feller. In any category there are the following fundamental problems concerning embeddings from an object Z into another object X: 1. (Existence) Does there exist an embedding of Z into X? 2. (Uniqueness) Having two embeddings f, g of Z into X, does there exists an automorphism ψ of X such that g = ψ ◦ f? In this talk, we will mainly focus on the first problem in the category of affine varieties, where Z is smooth and X is an algebraic group. Amongst other things, we will discuss the following result. Theorem. For every simple algebraic group G and every smooth affine variety Z with dim G > 2dim Z + 1, there exists an embedding of Z into G. The proof is based upon parametric transversality results for flexible affine varieties due to Kaliman. We will also discuss the following result, which implies the optimality of the above existence result up to a possible improvement of the dimension bound by one. It’s proof is an adaptation of a Chow-group-based argument due to Bloch, Murthy, and Szpiro for the affine space. Proposition. For every non-finite algebraic group G and every d ≥ dim G / 2, there exists an irreducible smooth affine variety of dimension d that does not admit an embedding into G.