Multiferroic nanostructures have been attracting tremendous attention over the past decade, due to their rich cross-coupling effects and prospective electronic applications. In particular, the emergence of some exotic phenomena in size-confined multiferroic systems, including topological domain states such as vortices, center domains, and skyrmion bubble domains, has opened a new avenue to a number of intriguing physical properties and functionalities, and thus underpins a wide range of applications in future nanoelectronic devices. It is also highly appreciated that nano-domain engineering provides a pathway to control the magnetoelectric properties, which is promising for future energy-efficient spintronic devices. In recent years, this field, still in its infancy, has witnessed a rapid development and a number of challenges too. In this article, we shall review the recent advances in the emergent domain-related exotic phenomena in multiferroic nanostructures. Specific attention is paid to the topological domain structures and related novel physical behaviors as well as the electric-field-driven magnetic switching via domain engineering. This review will end with a discussion of future challenges and potential directions.