Solving systems of linear equations and differential equations are fundamental tasks in scientific and engineering computation. While classical algorithms typically take time at least proportional to the number of variables, a quantum computer with suitable input access can produce a quantum state that encodes the problem solution with logarithmic complexity, and thus can potentially achieve an exponential speedup over a classical computer. In this talk, we will discuss high-level ideas and underlying mathematical structures of several quantum algorithms for linear systems and differential equations. For systems of linear equations, we will discuss an efficient quantum algorithm based on time-optimal discrete adiabatic evolution and the eigenvector filtering technique. For differential equations, we focus on the framework of simulating a general class of non-unitary dynamics as a linear combination of Hamiltonian simulation problems.