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We introduce an extension of cone splines and box splines to fractional andcomplex orders. These new families of multivariate splines are defined in theFourier domain along certain $s$-dimensional meshes and include as specialcases the three-directional box splines \cite{article:condat} and hex splines\cite{article:vandeville} previously considered by Condat, Van De Ville et al.These cone and hex splines of fractional and complex order generalize theunivariate fractional and complex B-splines defined in\cite{article:ub,article:fbu} and investigated in, e.g.,\cite{article:fm,article:mf}. Explicit time domain representations are derivedfor these splines on $3$-directional meshes. We present some properties ofthese two multivariate spline families such as recurrence, decay andrefinement. Finally it is shown that a bivariate hex spline and its integerlattice translates form a Riesz basis of its linear span.

Fractional Cone Splines and Hex Splines