Full vaccination status is already accepted by many countries as a means of gaining entry without the need for additional testing. Currently however, a valid, negative Covid test, (a PCR or rapid antigen test), is still required to enter certain destinations.
Due to the rapidly changing nature of the situation, please check the applicable rules with the authorities of the destination country in question.
Useful up-to-date information is available at: https://reopen.europa.eu/en.
Please refer to the rules of the individual countries in respect of vaccines and the appropriate proof of vaccination.