These days the various countries are launching their own strategies to open their national
borders.
Spain from 01.06.2021 opens its borders with the following indications:
– Citizens vaccinated with both doses of one of the vaccines recognized by the WHO or EMA will be able to enter if the second dose or single dose is performed, without the obligation of negative PCR;
– In the absence of this document, they must have with them a certificate of occurred PCR or antigenic swab (recognized by the European Union) within 48 hours prior the entrance
– Or the certificate of healing from COVID19 issued at least 11 days after the first positive swab and in any case issued in the 180 days prior to entry.

France from 09.06.2021 has opened its borders with the following indications:
– All travelers from “Green” countries (European Area, Australia, South Korea, Israel, Japan, Lebanon, New Zealand and Singapore) only if vaccinated with double dose since 14 days or 4 weeks with single dose vaccine
– Arrivals from South Africa, Argentina, Bahrain, Bangladesh, Bolivia, Brazil, Chile, Colombia, Costa Rica, India, Nepal, Pakistan, Sri Lanka, Suriname, Turkey and
Uruguay will only be able to enter for imperative reasons and must present a swab performed within 48 hours prior the entrance to France and upon arrival, they will be subjected to a rapid swab with the obligation of 7 days of quarantine
– For all other countries, it is mandatory to submit a PCR test performed within 72
hours prior to departure to France

Read the legislation
http://www.viaggiaresicuri.it/country/ESP
http://www.viaggiaresicuri.it/country/FRA