The Grassmann.jl package provides tools for doing computations based on multi-linear algebra, differential geometry, and spin groups using the extended tensor algebra known as Grassmann-Clifford-Hestenes-Taylor geometric algebra. The primary operations are ∧, ∨, ⋅, *, ×, ⋆, ', ~ (which are the outer, regressive, inner, geometric, and cross products along with the Hodge star, adjoint, and multivector reversal operations). Any operations are truly extensible with high dimensional support for up to 62 indices and staged caching / precompilation, where the code generation enables the fairly automated task of making more definitions. The DirectSum.jl multivector parametric type polymorphism is based on tangent bundle vector spaces and conformal projective geometry to make the dispatch highly extensible for many applications. Additionally, interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built.