Engineered transcriptional activator-like effectors (TALEs) are versatile tools for genome manipulation with applications in research and clinical contexts. One current drawback of TALEs is that the 5′ nucleotide of the target is specific for thymine (T). TALE domains with alternative 5′ nucleotide specificities could expand the scope of DNA target sequences that can be bound by TALEs. Another drawback of TALEs is their tendency to bind and cleave off-target sequence, which hampers their clinical application and renders applications requiring high-fidelity binding unfeasible. This disclosure provides methods and strategies for the continuous evolution of proteins comprising DNA-binding domains, e.g., TALE domains. In some aspects, this disclosure provides methods and strategies for evolving such proteins under positive selection for a desired DNA-binding activity and/or under negative selection against one or more undesired (e.g., off-target) DNA-binding activities. Some aspects of this disclosure provide engineered TALE domains and TALEs comprising such engineered domains, e.g., TALE nucleases (TALENs), TALE transcriptional activators, TALE transcriptional repressors, and TALE epigenetic modification enzymes, with altered 5′ nucleotide specificities of target sequences. Engineered TALEs that target ATM with greater specificity are also provided.