Recently Chen, [K. Chen, Signature with message recovery, Electronics Letters, 34(20) (1998) 1934], proposed a signature with message recovery. But Mitchell and Yeun [C. J. Mitchell and C. Y. Yeun, Comment - signature with message recovery, Electronics Letters, 35(3) (1999) 217] observed that Chen's scheme is only an authenticated encryption scheme and not a signature scheme as claimed. In this article, we propose a new signature scheme in the sense of Mitchell and Yeun and with message recovery feature. The designated verifier signature is introduced by Jakobsson et al. [M. Jakobsson, K. Sako, R. Impagliazzo, Designated verifier proofs and their applications, Proc. of Eurocrypt’96, LNCS 1070 (1996) pp. 143–154]. We propose a designated verifier signature scheme with non-repudiation of origin. We also give a protocol for a convertible designated verifier signature scheme with non-repudiation of origin. Both of these schemes are based on our proposed signature scheme with message recovery.