Methods and devices are provided for administering a drug to a patient's eye. The methods include (a) inserting a hollow microneedle into the sclera or corneal stroma without penetrating across the sclera or corneal stroma; and (b) infusing a fluid drug formulation through the microneedle and into the sclera or cornea. It further may include partially retracting the microneedle before infusion to enhance delivery. Alternatively, the methods may include (a) inserting a solid microneedle into the sclera or corneal stroma without penetrating across the sclera or corneal stroma, wherein the solid microneedle comprises a first quantity of a drug formulation and inserting causes the solid microneedle to form a pocket in the sclera or corneal stroma; and (b) releasing the drug formulation into the pocket to form a drug depot, whereby a drug is released from the depot. The methods and devices may include an array of multiple microneedles.