The authors define a supercurve as a generalization of a super Riemann surface, (SRS): specifically, an SRS is a supercurve that admits an odd global differential operator with nowhere vanishing square that takes values in some invertible sheaf. Supercurves are spaces equipped with sheaves of supercommutative \(\Lambda\)-algebras, where \(\Lambda\) is a (not necessarily reduced) Grassmann algebra over \(\mathbb{C}\). The motivation for the authors' definition is to produce a super Krichever map and obtain the flows of the super KP hierarchy defined by \textit{M. Mulase} and \textit{J. M. Rabin} [Int. J. Math. 2, No. 6, 741-760 (1991; Zbl 0754.58003)]. In the process, the authors develop an interesting theory of dual curves, Berezinian (dualizing sheaf for the dual curve), integration on supercontours, cohomology theory, period matrices, Poincaré sheaf and theta functions. They develop independently a notion of super Grassmannian and super Heisenberg algebra (which acts by flows on the super Grassmannian), with the analog of the KP equations for the \(\tau\)-function. Via a super Krichever map, finally, the authors construct algebro-geometric solutions in terms of super theta functions. As they point out, these are not like other super KP theories, for which the super curve varies as well as a point of the Jacobian [cf. e.g. \textit{Yu. I. Manin} and \textit{A. O. Radul}, Commun. Math. Phys. 98, 65-77 (1985; Zbl 0605.35075) and \textit{V. G. Kac} and \textit{J. W. van de Leur}, Infinite dimensional Lie algebras and groups, Proc. Conf., Marseille/Fr. 1988, Adv. Ser. Math. Phys. 7, 369-406 (1989; Zbl 0748.17021)].