Let O denote the set of non-degenerate, indefinite, real quadratic forms in 3-variables. We define for every such quadratic form Q, the Markov infimum m(Q)=inf{|Q(v)|^3/|det(Q)|: v is a nonzero integral vector in R^3}. This normalization makes the infimum invariant after rescaling the quadratic form. The set M={m(Q): Q in O} is called the 3-dimensional Markov spectrum. An early result of Cassels-Swinnerton-Dyer combined with Margulis’ proof of the Oppenheim conjecture asserts that M consists of rational numbers, and for every a0 there are only finitely many numbers in M which are greater than a. In this lecture we will discuss an effective improvement of this result. This is an ongoing joint work with Prof. Margulis. The key ingredient is to study the compact orbits of the SO(2,1) action on SL(3, R)/SL(3, Z), and our method involves techniques from the geometry of numbers, dynamics on homogeneous spaces and automorphic representations.