Scattered radiation remains one of the primary challenges for digital mammography, resulting in decreased image contrast and visualization of key features. While anti-scatter grids are commonly used to reduce scattered radiation in digital mammography, they are an incomplete solution that can add radiation dose, cost, and complexity. Instead, a software-based scatter correction method utilizing asymmetric scatter kernels is developed and evaluated in this work, which improves upon conventional symmetric kernels by adapting to local variations in object thickness and attenuation that result from the heterogeneous nature of breast tissue. This fast adaptive scatter kernel superposition (fASKS) method was applied to mammography by generating scatter kernels specific to the object size, x-ray energy, and system geometry of the projection data. The method was first validated with Monte Carlo simulation of a statistically-defined digital breast phantom, which was followed by initial validation on phantom studies conducted on a clinical mammography system. Results from the Monte Carlo simulation demonstrate excellent agreement between the estimated and true scatter signal, resulting in accurate scatter correction and recovery of 87% of the image contrast originally lost to scatter. Additionally, the asymmetric kernel provided more accurate scatter correction than the conventional symmetric kernel, especially at the edge of the breast. Results from the phantom studies on a clinical system further validate the ability of the asymmetric kernel correction method to accurately subtract the scatter signal and improve image quality. In conclusion, software-based scatter correction for mammography is a promising alternative to hardware-based approaches such as anti-scatter grids.