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We study the representation theory of the superconformal algebra$W_k(g,f_{\theta})$ associated with a minimal gradation of $g$. Here, $g$ is asimple finite-dimensional Lie superalgebra with a non-degenerate, evensupersymmetric invariant bilinear form. Thus, $W_k(g,f_{\theta})$ can be one ofthe well-known superconformal algebras including the Virasoro algebra, theBershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the Bershadsky-Knizhnikalgebras, the N=2 superconformal algebra, the N=4 superconformal algebra, theN=3 superconformal algebra and the big N=4 superconformal algebra. We prove the conjecture of V. G. Kac, S.-S. Roan and M. Wakimoto for$W_k(g,f_{\theta})$. In fact, we show that any irreducible highest weightcharacter of $W_k(g,f_{\theta})$ at any level $k\in C$ is determined by thecorresponding irreducible highest weight character of the Kac-Moodyaffinization of $g$.

Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture