What are the connections between Portal series plot and Half-Life series plot?

Aperture Science (which is the firm that developed the Portal gun) is in direct competition with Black Mesa (which is the firm for which Gordon Freeman worked) for military contracts

See the section Other facts on this link

Aperture Science and Black Mesa were bitter rivals. As seen in the Portal maps "testchmb_a_15" and "escape_02", Aperture Science employees were briefed on Black Mesa through slideshow presentations, such as one titled "Dollar$ and Sense: Competing with Black Mesa for DoD and Government-wide Acquisition Contracts"

EDIT: Here's an excellent guide on Half Life: http://members.shaw.ca/halflifestory/ Check out the timeline for an awesome detail of all events in the Half Life saga, and in the bottom a little speculation to its relation to Portal:

The date that the events of Portal take place aren't certain. We do know that GLaDOS says to Chell: "Are you trying to escape? [chuckle] Things have changed since the last time you left the building. What's going on out will make you wish you were back in here. I have an infinite capacity for knowledge, and even I'm not sure what is going on outside. All I know is I'm the only thing standing between us and them. Well, I was. Unless you have a plan for building some supercomputer parts in a big hurry, this place isn't going to be safe much longer." This implies that the game takes place during the Portal Storms or afterwards.

This site is not official but it's a great analysis IMO.

• In the original Portal, just before the chamber containing GlaDOS, there is a projector with some slides detailing that Black Mesa and Aperture Science are in direct competition.

• In the song at the end of Portal, GlaDOS mentions Black Mesa: "Maybe you'll find someone else to help you. Maybe Black Mesa... that was a joke, ha-ha, fat chance"

• In HL2:Ep2, there are references to the Borealis, an Aperture Science research boat that went missing.

  • In Portal 2 (which takes place in an Aperture facility), they mention a boat-yard where some sort of "time-travel experiment" went wrong. Later, you find an empty ship-port, with life-savers scattered around that say "Borealis."

  • This concept art for HL2:Ep3 contains the Borealis.

• In Portal 2, Cave Johnson (CEO of Aperture Science) states his hate towards Black Mesa in a recording, claiming they "stole their technology" from Aperture Science.

• In Portal 2, (just before you are first introduced to the goo) there is a room in Aperture Science with a bunch of trophies and awards for "#2 Applied-Science company of the year," implying (along with the projector slides from Portal 1) that the #1 company is Black Mesa.

This page has a timeline of the rivalry between Black Mesa and Aperture Science

In HL2 Ep 2, You are told about an aperture ship called Borealis, it is suggested that this ship will appear in episode 3.

So both games take place in the same universe, and Valve seem to be intertwining the two stories. There has also been rumour of a portal gun appearing in episode 3