A combustor assembly for a turbine comprising: a combustor including a combustor liner; a first flow sleeve surrounding said combustor liner forming a first substantially axially-extending flow annulus radially therebetween, said first flow sleeve having a first plurality of apertures formed about a circumference thereof for directing compressor discharge air as cooling air radially into said first flow annulus; a transition piece connected to said combustor liner, said transition piece adapted to carry hot combustion gases to the turbine; a second flow sleeve surrounding said transition piece forming a second substantially axially-extending flow annulus radially therebetween, said second flow sleeve having a second plurality of apertures for directing compressor discharge air as cooling air radially into said second flow annulus, said first substantially axially-extending flow annulus connecting with said second substantially axially-extending flow annulus; a resilient annular seal structure disposed radially between an aft end of said combustor liner and a forward end of said transition piece, said resilient annular seal structure configured to form a first annular cavity radially between said forward end of said transition piece and said aft end of said combustor liner and a second annular cavity formed between the aft end portion of the combustor liner and an annular cover plate, the aft end portion of the annular cover plate formed with a plurality of cooling air exit holes for supplying cooling air from the first annular cavity to the second annular cavity; the combustor assembly being characterized in that it further comprises at least one transfer tube radially extending from said second flow sleeve through said second flow annulus to said transition piece, and arranged to supply compressor discharge cooling air radially from an area outside said first and second substantially axially-extending flow annuli (64, 60) directly to said resilient annular seal structure and to said aft end of said combustor liner wherein said forward end of said transition piece is formed with a first annular plenum chamber, and wherein, in use, said at least one transfer tube supplies compressor discharge cooling air to said first annular plenum chamber which, in turn, supplies the compressor discharge cooling air to said resilient annular seal structure and to said aft end of said combustor liner, and wherein said first annular plenum chamber is provided with plural, circumferentially-spaced cooling air exit apertures provided in the radially-inner wall portion of the transition piece and substantially radially aligned with said resilient annular seal structure. a combustor including a combustor liner; a first flow sleeve surrounding said combustor liner forming a first substantially axially-extending flow annulus radially therebetween, said first flow sleeve having a first plurality of apertures formed about a circumference thereof for directing compressor discharge air as cooling air radially into said first flow annulus; a transition piece connected to said combustor liner, said transition piece adapted to carry hot combustion gases to the turbine; a second flow sleeve surrounding said transition piece forming a second substantially axially-extending flow annulus radially therebetween, said second flow sleeve having a second plurality of apertures for directing compressor discharge air as cooling air radially into said second flow annulus, said first substantially axially-extending flow annulus connecting with said second substantially axially-extending flow annulus; a resilient annular seal structure disposed radially between an aft end of said combustor liner and a forward end of said transition piece, said resilient annular seal structure configured to form a first annular cavity radially between said forward end of said transition piece and said aft end of said combustor liner and a second annular cavity formed between the aft end portion of the combustor liner and an annular cover plate, the aft end portion of the annular cover plate formed with a plurality of cooling air exit holes for supplying cooling air from the first annular cavity to the second annular cavity; the combustor assembly being characterized in that it further comprises at least one transfer tube radially extending from said second flow sleeve through said second flow annulus to said transition piece, and arranged to supply compressor discharge cooling air radially from an area outside said first and second substantially axially-extending flow annuli (64, 60) directly to said resilient annular seal structure and to said aft end of said combustor liner wherein said forward end of said transition piece is formed with a first annular plenum chamber, and wherein, in use, said at least one transfer tube supplies compressor discharge cooling air to said first annular plenum chamber which, in turn, supplies the compressor discharge cooling air to said resilient annular seal structure and to said aft end of said combustor liner, and wherein said first annular plenum chamber is provided with plural, circumferentially-spaced cooling air exit apertures provided in the radially-inner wall portion of the transition piece and substantially radially aligned with said resilient annular seal structure.