Nanoparticles are a state of matter that has properties different from either molecules or bulk solids. In the present work, we review the shape and structure of nanometer-sized particles; several shapes are discussed, such as the octahedron and truncated octahedron, the icosahedron, the Marks decahedron, the truncated "star-like" decahedron, the rounded decahedron and the regular decahedron. Experimental high-resolution transmission electron microscopy (TEM) images of each type of particle are presented together with the Fast Fourier Transform and a model of the particle. We consider only gold particles grown by vapor deposition or by colloidal methods. High-resolution TEM images of the particles in different orientations are shown. We discuss two basic types of particles uncapped and capped. Data for other metals and semiconductors are reviewed. We have also performed very extensive simulations obtaining the total energy and pair correlation functions for each cluster under study. Furthermore, distributions of single atom energy for every cluster are displayed in order to reveal the effect of surface on the stability of different types and sizes of clusters. We discuss the structure of the particles from similar to1 to similar to 100 nm. The mechanisms for stress release as the particles grow larger are reviewed and a mechanism is suggested. Finally, we discuss the parameters that define the shape of a nanoparticle and the possible implications in technological applications.