Spatial organization of protein biosynthesis in the eukaryotic cell has been studied for more than fifty years, thus many facts have already been included in textbooks. According to the classical view, mRNA transcripts encoding secreted and transmembrane proteins are translated by ribosomes associated with endoplasmic reticulum membranes, while soluble cytoplasmic proteins are synthesized on free polysomes. However, in the last few years, new data has emerged, revealing selective translation of mRNA on mitochondria and plastids, in proximity to peroxisomes and endosomes, in various granules and at the cytoskeleton (actin network, vimentin intermediate filaments, microtubules and centrosomes). There are also long-standing debates about the possibility of protein synthesis in the nucleus. Localized translation can be determined by targeting signals in the synthesized protein, nucleotide sequences in the mRNA itself, or both. With RNA-binding proteins, many transcripts can be assembled into specific RNA condensates and form RNP particles, which may be transported by molecular motors to the sites of active translation, form granules and provoke liquid-liquid phase separation in the cytoplasm, both under normal conditions and during cell stress. The translation of some mRNAs occurs in specialized "translation factories," assemblysomes, transperons and other structures necessary for the correct folding of proteins, interaction with functional partners and formation of oligomeric complexes. Intracellular localization of mRNA has a significant impact on the efficiency of its translation and presumably determines its response to cellular stress. Compartmentalization of mRNAs and the translation machinery also plays an important role in viral infections. Many viruses provoke the formation of specific intracellular structures, virus factories, for the production of their proteins. Here we review the current concepts of the molecular mechanisms of transport, selective localization and local translation of cellular and viral mRNAs, their effects on protein targeting and topogenesis, and on the regulation of protein biosynthesis in different compartments of the eukaryotic cell. Special attention is paid to new systems biology approaches, providing new cues to the study of localized translation.