In 1970 Matiyasevich, building on work by Davis, Putnam and Robinson, proved that Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by considering polynomial equations over commutative rings other than the integers. The biggest open problem in the area is Hilbert's Tenth Problem over the rational numbers. In this talk we will construct some subrings $R$ of the rationals that have the property that Hilbert's Tenth Problem for $R$ is Turing equivalent to Hilbert's Tenth Problem over the rationals. We will also discuss some recent undecidability results for function fields of positive characteristic.