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We discuss the current state ofproblems associated with the discovery and study ofexocomets, i.e objects of extrasolar systems, that are inmany aspects similar to the solar system comets. Thusfar, more than 4,300 exoplanetary systems have beendiscovered, but little is known about the populationsof subplanetary bodies in these systems, in particularcomets. Existing theories of planet formation suggestthat populations of small bodies in the planetarysystems should be numerous, especially at the earlystages of their formation. Currently, most of thestars observed with the confirmed transit exoplanetsand candidate stars with exoplanetary systems havebeen collected in the Kepler and TESS space mi-ssion databases. These databases can be used for asearch and study of the exocomet transit signaturesin extrasolar systems. So far, the number of observedcases of exocomet transits has been small, only about20 events. Due to the rapid accumulation of data, newresearches aimed to identify the specific transit eventsand study the physical characteristics of small bodies inthe extrasolar systems are of great importance.Worthconsidering is the concept of “falling evaporatingbodies”, massive enough bodies surrounded by mini-atmospheres, which fall on the parent star. The eventsmight produce the variable, short-time red-shifts ofsome spectral features in their spectra. Alternativeevidence of exocomet transits can be obtained bydetection of the cometary emissions in CO, C, andO lines in the millimeter region of the spectra ofdebris disks by the ALMA and APEX telescopes.We discuss the known detection methods based onthe analysis of the photometric and spectral seriesof observational data of space missions and ground-based complexes. Some results of experimental studi-es of the exocometary transit signatures obtained byother authors are presented and discussed.

SIGNATURES OF EXOCOMET ACTIVITY AROUND STARS AT DIFFERENT EVOLUTIONARY STAGES