Tissue engineering has emerged as an important research area that provides numerous research tools for the fabrication of biologically functional constructs that can be used in drug discovery, disease modeling, and the treatment of diseased or injured organs. From a materials point of view, scaffolds have become an important part of tissue engineering activities and are usually used to form an environment supporting cellular growth, differentiation, and maturation. Among various materials used as scaffolds, hydrogels based on natural polymers are considered one of the most suitable groups of materials for creating tissue engineering scaffolds. Natural hydrogels, however, do not always provide the physicochemical and biological characteristics and properties required for optimal cell growth. This review discusses the properties and tissue engineering applications of widely used natural hydrogels. In addition, methods of modulation of their physicochemical and biological properties using soft nanoparticles as fillers or reinforcing agents are presented.