Google appealed against a record 2.4-billion-euro ($2.9 billion) EU antitrust fine on Monday, while Facebook has been fined 1.2 million euros ($1.4 million) for allegedly collecting personal information from users in Spain.
In Google's case, the European regulator had ruled that positioning its own shopping comparison service at the top of Google search results was an abuse of power.
The fine was the largest penalty ever issued by the regulator, which also said the firm could face more fines if it continued its practices.
At the time that the fine was imposed, Margrethe Vestager, the European Union's Competition Commissioner said that Google's activity was "illegal under EU antitrust rules".
Google remains "respectfully disagreed" with the ruling.
Google was also given 90 days to end the "anti competitive" practices or face a further fine amounting to 5% of the average daily global earning of its parent company Alphabet.
The deadline for making the changes is 28 September.
The Luxembourg-based General Court, Europe's second-highest, is expected to take years before ruling on Google's case.
Last week, the EU Court of Justice (ECJ) told a lower tribunal to re-examine U.S. chipmaker Intel's appeal against a 1.06 billion euro fine, dealing a rare setback to the Commission.
While the Google case is not the same as that of Intel, the judgment has been welcomed by companies under EU scrutiny because it raises the bar for the regulator to prove wrongdoing.
In a sepaerate case, Facebook has been fined 1.2 million euros ($1.4 million) for allegedly collecting personal information from users in Spain that could then be used for advertising, the national data protection watchdog said on Monday.
The fine stemmed from an investigation into the social network company conducted alongside similar probes in Belgium, France, Germany and the Netherlands, the AEPD authority said.
The 1.2 million euro fine is a fraction of Facebook's quarterly revenue of about $8 billion and stock market capitalization of around $435 billion.
AEPD said it found three cases in which Facebook had collected details such as the gender, religious beliefs, personal tastes and browsing history of its millions of Spanish users without informing them how such information would be used.
According to the AEPD, the tech giant did not inform users about how it would use data collected on third-party websites, and did not obtain consent to use it.
Using cookies, Facebook also collects data from people who do not have an account on the social network but navigate other pages containing a "like" button, AEPD said.