Application of semiconductors as electric and optoelectronic devices depends critically on their dopability. Failure to dope a material, i.e., to produce enough free charge carriers beyond a certain limit, is often the single most important bottleneck for advancing semiconductor-based high technology. Using the first-principles band structure method, we have studied systematically the general chemical trends of the defect formation and ionization in semiconductors to understand the physical origin of the doping difficulty. New approaches to overcoming the doping limit have been developed. This paper reviews our recent progress and discusses some of the computational issues in defect calculations.