The development of the theory of classical Jacobi theta functions and their connections to elliptic integrals was a crowning achievement of Nineteenth Century mathematics. In \textit{S. Ramanujan}'s Notebooks, and in his 1914 paper on series representations for \(1/\pi\) [Quart. J. 45, 350--372 (1914; JFM 45.1249.01)], he refers to extensions of classical results corresponding to alternative theories of elliptic functions. Ramanujan's work was proved, interpreted and extended beginning in the latter half of the same century through the work of J. M. and P. B. Borwein, M. Hirschorn, F. Garvan, B. C. Bernt, S. Bhargava, and others. In the current work, a comprehensive set of results is developed for a cubic analogue of the classical Jacobi theta function \[ \theta_{3}(z\mid q) = \sum_{n=-\infty}^{\infty} q^{n{^2}} z^{n} \] defined by \[ \Theta_{3}(z_{1}, z_{2} \mid q)\sum_{m,n=-\infty}^{\infty} q^{m^{2}+mn+n^2}z_{1}^{m}z_{2}^{n}. \] The theory of the cubic theta function \(\Theta_{3}(z_{1}, z_{2} \mid q)\) is closely linked to the genus 2 hyperelliptic curve \[ y^{3} = x(1-x)^{2} (1 - mx), \] in the same way the classical theta function \(\theta_{3}(z \mid q)\) is related to the elliptic curve \[ y^{2} = x(1 - x)(1 - mx). \] A theory of four allied cubic theta functions is developed in analogue to the theory of the four Jacobi theta functions, including formulation of identities involving corresponding vector-valued analogues of the classical elliptic integrals. Inversion formula for the cubic theta functions are derived. Examples of addition theorems for the cubic theta function are formulated, and a resulting series-to-product identity is derived. Generalized modular transformations are formulated that extend known cubic modular equations. In this paper, D. Schultz fills a substantial gap in the development of the cubic theory of elliptic functions and leads the reader through a full realization of Ramanujan's vision.