LEFTfield forward models cosmological matter density fields and biased tracers of large-scale structure. The model, written in C++ code, is centered around classes encapsulating scalar, vector, and tensor grids. It includes the complete bias expansion at any order in perturbations and captures general expansion histories without relying on the EdS approximation; however, the latter is also implemented and results in substantially smaller computational demands. LEFTfield includes a subset of the nonlinear higher-derivative terms in the bias expansion of general tracers.