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In this paper, a finite volume element (FVE) method is considered for spatial approximations of time-fractional diffusion equations involving a Riemann-Liouville fractional derivative of order $\alpha \in (0,1)$ in time. Improving upon earlier results (Karaa {\it et al.}, IMA J. Numer. Anal. 2016), optimal error estimates in $L^2(\Omega)$- and $H^1(\Omega)$-norms for the semidiscrete problem with smooth and middly smooth initial data, i.e., $v\in H^2(\Omega)\cap H^1_0(\Omega)$ and $v\in H^1_0(\Omega)$ are established. For nonsmooth data, that is, $v\in L^2(\Omega)$, the optimal $L^2(\Omega)$-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the $L^\infty(\Omega)$-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time-fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.