The goal of the MAGMA project is to create a new generation of linear algebra libraries that achieves the fastest possible time to an accurate solution on heterogeneous architectures, starting with current multicore + multi-GPU systems. To address the complex challenges stemming from these systems' heterogeneity, massive parallelism, and the gap between compute speed and CPU-GPU communication speed, MAGMA's research is based on the idea that optimal software solutions will themselves have to hybridize, combining the strengths of different algorithms within a single framework. Building on this idea, the goal is to design linear algebra algorithms and frameworks for hybrid multicore and multi-GPU systems that can enable applications to fully exploit the power that each of the hybrid components offers.