The course will cover some basic multiscale methods as well as some advanced methods for solving partial differential equations with multiple scales. The topics will include: Background, Problems with multiple scales; Difficulties in solving multiscale problems; Homogenization techniques for partial differential equations (PDEs) (with periodic micro-structure); Formal asymptotic analysis; Homogenized media properties. Applications to various PDEs: Effective medium theory (based on homogenization); Simplified theories; Bounds for homogenized coefficients: Numerical homogenization (upscaling) techniques; Slowly varying and non-periodic microstructures; Estimating errors of numerical homogenization: Homogenization for nonlinear operators; Numerical homogenization for nonlinear operators; Multiscale finite element methods; Differences from homogenization/numerical homogenization; Simplified multiscale basis functions.