We review methods for validated arbitrary-precision numerical computation of elliptic functions and their inverses (the complete and incomplete elliptic integrals), as well as the closely related Jacobi theta functions and $\mathrm{SL}_2(\mathbb{Z})$ modular forms. A general strategy consists of two stages: first, using functional equations to reduce the function arguments to a smaller domain; second, evaluation of a suitable truncated series expansion. For elliptic functions and modular forms, one exploits periodicity and modular transformations for argument reduction, after which the rapidly convergent series expansions of Jacobi theta functions can be employed. For elliptic integrals, a comprehensive strategy pioneered by B. Carlson consists of using symmetric forms to unify and simplify both the argument reduction formulas and the series expansions (which involve multivariate hypergeometric functions). Among other aspects, we discuss error bounds as well as strategies for argument reduction and series evaluation that reduce the computational complexity. The functions have been implemented in arbitrary-precision complex interval arithmetic as part of the Arb library.