We look back at the province's previous Heineken Cup appearances
A few weeks ago there may well have been a bit of worry about Connacht heading into the European Champions Cup.
They had started the Guinness Pro 12 season poorly, failing to live up to the performances that had seen them become neutral's favourites en route to becoming champions for the very first time last season.
But as the European Champions Cup challenge of French side Toulouse awaits at the Sportsground at 5.35pm on Saturday, there is a bit more optimism out west.
Consecutive wins over Ulster and Edinburgh not only means precious points on the board but also boosts domestic confidence that can be transferred onto the European platform.
So much so indeed, that ex-Ireland captain Keith Wood feels Connacht have a chance against Toulouse.
On Wednesday he told Off The Ball, that "The fact and the manner in which they've got it right over the last two weekends has been phenomenal actually and their performance last weekend was pretty incredible."
And he also concluded that he "would hold out an element of optimism for Connacht this weekend."
But let's not forget that Connacht have been at Europe's top table before - albeit not at the current heights they have reached since Pat Lam became head coach.
This season isn't Connacht's first at the highest level and also not the first time they have run into Toulouse.
In the 2011-12 Heineken Cup, Connacht qualified for the pool stage because then-European champions Leinster had already qualified domestically, leaving an extra spot for an Irish side.
Under Eric Elwood's stewardship at the time, the province ended up pooled with Harlequins, Gloucester as well as Toulouse.
And it wasn't a success on the results board with five defeats from six.
But they did manage one victory with three Niall O'Connor penalties kicking them to a 9-8 victory over Gloucester in their final group game.
Incidentally they lost home and away Toulouse by a 60-13 score.
Connacht's Gavin Duffy and Denis Buckley after the 2013 Toulouse match ©INPHO/Billy Stickland
With Leinster winning Europe's top prize again in 2012, Connacht returned for another season in the Heineken Cup pool stages.
This was the start of the Toulon era of dominance. But Connacht, who were now in the Tier three pot, were in a pool that featured Biarritz, Harlequins and Zebre.
And it was a far better showing second time out as they finished on 12 points, beating Zebre home and away as well as Biarritz at home.
Indeed, they fell three points short of finishing second in the pool with Biarritz taking that spot, although the French side were not one of the best second placed sides so missed out on the quarter-finals.
A Tier 3 side again the pool draw, they were again placed in the same pool as Zebre, while the two strongest sides were Toulouse and Saracens.
Pat Lam had become head coach by this point and like the previous campaign, defeated Zebre home and away.
And in hopefully what is an omen against Toulouse this Satuday, they defeated the French side 16-14 in a historic result at Stade Ernest-Wallon, with Kieran Marmion's try ably aided by kicks from Dan Parks.
However, Connacht did lose at home to Toulouse and also suffered a massive 64-6 loss to Saracens in their final pool match, conceding 11 tries in the process.
Ultimately Lam's team finished third in the pool with 13 points.
But the Toulouse win remains their greatest Heineken Cup/Champions Cup triumph to date.