In this talk, we present a brief overview on the geometrical structure of Laplacian eigenfunctions. In particular, we discuss the problem of localization when an eigenfunction essentially ``lives'' on a small subset of the domain and takes small values on the remaining part. In the high-frequency limit, this is a classical problem in the theory of quantum billiards. However, less attention has been paid to localization of low-frequency eigenfunctions. We present several recent results on the exponentialdecay of low-frequency eigenfunctions. \ \ References: \ 1) Delitsyn, Nguyen, Grebenkov, Exponential decay of Laplacian eigenfunctions in domains with branches of variable cross-sectional profiles, Eur. Phys. J. B 85, 371 (2012). \ 2) Delitsyn, Nguyen, Grebenkov, Trapped modes in finite quantum waveguides, Eur. Phys. J. B 85, 176 (2012). \ 3) Grebenkov, Nguyen, Geometrical Structure of Laplacian Eigenfunctions (SIAM Reviews), online http://arxiv.org/abs/1206.1278