Britain is due to leave the European Union on Friday if an extension is not granted.
But how will events this week decide what happens?
Tuesday April 9
She is expected to set out the rationale behind her request for a delay to the date of Brexit from April 12 to June 30.
The trip appears to make it unlikely that any compromise Brexit deal thrashed out with Labour will be put before MPs ahead of the summit.
Wednesday April 10
The Prime Minister will head to Brussels on Wednesday afternoon for an emergency meeting of the European Council.
She will ask the leaders of the 27 remaining EU nations to grant her an extension to Article 50 until June 30, without which the UK will leave with no-deal on Friday.
Thursday April 11
If the Cooper Bill has been passed by the Lords, it would place new requirements on the PM.
If the European Council proposes a different extension date, Mrs May would need to return to the Commons to obtain MPs’ approval.
It is also the final date for the UK to take steps to enable European Parliament elections to take place on May 23.
Friday April 12
This is when the UK is scheduled to leave the EU after MPs repeatedly rejected the Prime Minister’s deal.
Mrs May has written to European Council president Donald Tusk asking for a further extension to June 30, but the EU 27 will have to agree to it when they meet on Wednesday.
If they do not agree to an extension, Britain will leave without a deal at 11pm on Friday.