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In this talk we will present recent developments in the design and analysis of numerical homogenization methods. Numerical methods for linear and nonlinear partial differential equations that combine multiscale methods with reduced order modeling techniques such as the reduced basis method will be discussed. The talk is based upon a series of joint works with various collaborators[1,2,3,4,5]. [1] A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys., vol. 191, num. 1, p. 18-39, 2012. [2] A. Abdulle and Y. Bai, Reduced order modelling numerical homogenization, Philosophical Transactions of the Royal Society A, vol. 372, num. 2021, 2014. [3] A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Discrete Contin. Dyn. Syst. vol.8, num. 1, 2015. [4] A. Abdulle and O. Budac, An adaptive finite element heterogeneous multiscale method for Stokes flow in porous media, to appear in SIAM MMS. [5] A. Abdulle and P. Henning A reduced basis localized orthogonal decomposition, preprint submitted for publication. 2.2 A nonintrusive Reduced Basis Method applied to aeroacoustic simulations Fabien Casenave, Alexandre Ern Université Paris-Est, CERMICS (ENPC), 6 & 8 av Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France Tony Lelièvre Université Paris-Est, CERMICS (ENPC), 6 & 8 av Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France, and INRIA Rocquencourt, Matherials Team-Project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France