Since the afternoon hours today, PlayStation users trying to utilize the PSN services have experienced various difficulties even up to this hour. Many gamers in Europe and North America started experiencing slow download speeds on PSN. As expected, the company is aware of the issue.
We are aware that you may be experiencing issues with PSN. Thanks for your patience as we investigate. https://t.co/SrkLA6RbZz
— Ask PlayStation (@AskPlayStation) June 27, 2015
As of 7pm Pacific (10pm Eastern), PS4 users are having problems accessing the Account Management, Gaming and Social, PlayStation Now, PlayStation Vue and PlayStation Store. For PS3 users, the issues only affect accessing PlayStation Vue and PlayStation Store.
This issue is happening days before the expected routine PSN maintenance on Monday, June 29 from 9:30pm-11:00pm Pacific time.

[Update]: As of 11pm EST, the service has returned to normal, but Sony still reminds people about Monday’s routine PSN maintenance.
[Source]: PlayStation: PlayStation® Network PlayStation® Service Status & Ps4 Download speed.