Let β = β(2d) = 〖〖{β〗_i}〗_(iϵZ_(+ )^n ), with |i|≤ 2d, denote a real n-dimensional multisequence of degree 2d. For example, if n = 2 and d = 1, β(2) = {β00, β10, β01, β20, β11, β02}. For a closed set K in R^n, the truncated K-moment problem asks for conditions on β so that there exists a positive Borel measure μ on R^n, supported in K, such that β_i= ∫_(R^n)▒〖x^i dμ(x),〗 |i|≤ 2d. Concrete solutions are known only for a few sets K including , [0, +∞), [a,b], or when K is a planar curve of degree at most 2. A general solution, proved in collaboration with R.E. Curto, shows that β(2d) admits a K-representing measure if and only if the associated Riesz functional (Lβ(xi) = βi) admits a K-positive extension to polynomials of degree 2d+2. The intricate structure of positive polynomials causes difficulties in establishing K-positivity (so as to apply the extension theorem). We discuss some recent results with Jiawang Nie which resolve the truncated moment problem concretely in some previously open cases, including the case of the bivariate quartic problem (n = d = 2, K = R^2).