A simply connected solvable Lie group R R together with a left-invariant Riemannian metric g g is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds ( R , g ) (R,\,g) and ( R ′ , g ′ ) (R’ ,\,g’ ) may be isometric even when R R and R ′ R’ are not isomorphic. This article addresses the problems of (i) finding the "nicest" realization ( R , g ) (R,\,g) of a given solvmanifold, (ii) describing the embedding of R R in the full isometry group I ( R , g ) I(R,\,g) , and (iii) testing whether two given solvmanifolds are isometric. The paper also classifies all connected transitive groups of isometries of symmetric spaces of noncompact type and partially describes the transitive subgroups of I ( R , g ) I(R,\,g) for arbitrary solvmanifolds ( R , g ) (R,\,g) .