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Trapped samples of ultracold molecules are often short-lived, because closecollisions between them result in trap loss. We investigate the use ofshielding with static electric fields to create repulsive barriers betweenpolar molecules to prevent such loss. Shielding is very effective even forRbCs, with a relatively low dipole moment, and even more effective formolecules such as NaK, NaRb and NaCs, with progressively larger dipoles.Varying the electric field allows substantial control over the scatteringlength, which will be crucial for the stability or collapse of molecularBose-Einstein condensates. This arises because the dipole-dipole interactioncreates a long-range attraction that is tunable with electric field. For RbCs,the scattering length is positive across the range where shielding iseffective, because the repulsion responsible for shielding dominates. For NaK,the scattering length can be tuned across zero to negative values. For NaRb andNaCs, the attraction is strong enough to support tetraatomic bound states, andthe scattering length passes through resonant poles where these states crossthreshold. For KAg and CsAg, there are multiple bound states and multiplepoles. For each molecule, we calculate the variation of scattering length withfield and comment on the possibilities for exploring new physics.

Controlling collisional loss and scattering lengths of ultracold dipolar molecules with static electric fields