The "Capelli problem" for the symmetric pairs (gl × gl, gl) (gl, o), and (gl, sp) is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see [12], [15], [14], [18]). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs (g, k) := (gl(m|2n), osp(m|2n)) and (gl(m|n) × gl(m|n), gl(m|n)), acting on W := S 2 (C m|2n ) and C m|n ⊗ (C m|n ) * .To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra g maps surjectively onto the algebra PD(W ) g of g-invariant differential operators on the superspace W , thereby providing an affirmative answer to the "abstract" Capelli problem for W . Our proof works more generally for gl(m|n) acting on S 2 (C m|n ) and is new even for the "ordinary" cases (m = 0 or n = 0) considered by Howe and Umeda in [9].We next describe a natural basis {D λ } of PD(W ) g , that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi [12], [15], [20] by showing that the spectrum of D λ is given by a polynomial c λ , which is characterized uniquely by certain vanishing and symmetry properties.We further show that the top homogeneous parts of the eigenvalue polynomials c λ coincide with the spherical polynomials d λ , which arise as radial parts of k-spherical vectors of finite dimensional g-modules, and which are super-analogues of Jack polynomials. This generalizes results of Knop and Sahi [14].Finally, we make a precise connection between the polynomials c λ and the shifted super Jack polynomials of Sergeev and Veselov [25] for special values of the parameter. We show that the two families are related by a change of coordinates that we call the "Frobenius transform".