Proceeding from a theorem of Ueda which establishes an isomorphism (as modules for appropriate Hecke algebras) between certain spaces of cusp forms of integral and half-integral weight, the authors decompose the (full) space of newforms of half-integral weight into a direct sum of spaces of newforms of integral weight which occur with multiplicity one or two. This not only demonstrates in a precise way the failure of a multiplicity-one result to hold for half-integral weight newforms, but moreover indicates which spaces occur with a given multiplicity. The spaces occuring with multiplicity two are shown to be in one-to-one correspondence with a collection of Kohnen subspaces. Furthermore, it is shown under the Shimura correspondence that the level of a newform of half-integral weight is not determined by the level of the integral weight newform to which it corresponds; this is done by demonstrating the existence of half-integral weight newforms of arbitrarily high levels having the same eigenvalues as a given integral weight newform. This contrasts sharply with expectations arising from Waldspurger's results which characterize the square of the value of the squarefree Fourier coefficients of such half-integral weight forms in terms of a special value of twists of the \(L\)-series attached to the integral weight newforms. Finally, since the Waldspurger result does not give the sign of the Fourier coefficient, and for a given cusp form, there are necessarily an infinite number of choices of sign to be made, sufficient conditions regarding a knowledge of the squarefree coefficients are developed which allow half-integral weight newforms to be classified up to constant multiple. These conditions involve the development and study of an Atkin- Lehner type involution. In the last section, these later results are carried over to the Hilbert modular setting.