According to AIR FORCE Magazine (March 1995, p.61) Russian DA has only five Tu-160. Remaining 20 operational bombers were based in Ukraine.

From Open Media Research Institute:
93-03-25; UKRAINE ON STRATEGIC BOMBERS

In 1995 a deal was annonced: Ukraine will pay with Blackjacks and Bears for russian oil bill. Remaining Tu-160 and more than 30 Tu-95MS will be returned to Russia:
95-02-23; RUSSIA TO BUY UKRAINIAN BOMBERS
95-03-16; UKRAINE TO SELL BOMBERS TO RUSSIA AT BARGAIN PRICE

From: ssampson@icon.net (Steve Sampson)

In article <3tn9ma$ocj@newsbf02.news.aol.com> kavanaghny@aol.com 
(KavanaghNY) writes:

>       How does the Russian 'Blackjack' bomber compare to the B-1? How
>many 'Blackjacks' were built?.....does anyone one no of any web pages, ftp
>sites or books where I can find more info on the 'Blackjack'? and one more
>thing; Tu-???.

The Blackjack does not have even half the bomb load capability of the B-1.
Check Janes out.  It is real shocking how limited the payload is.  But I must
say, that the presence of the plane made Iceland Defense Forces (IDF) obsolete.
The plane could be within nuclear missle range (even without cutting across
Norway) faster than the E-3 and F-15's could scramble and intercept it.
Now that's power projection!  Thankfully we still only saw Bear, Bison and May
aircraft in the MADIZ.  The May (probably akin to the P-3) was real good at
laying down chaff corridors.  We thought it was a big weather system before
finally figuring it out :-)

From: amraam@netcom.com (Robin J. Lee)

Incidentally, the Blackjack is notorious for being unreliable and 
generally defective.  The two that they scraped up as display craft for 
Secretary Carlucci's visit to the Soviet Union in 1988 both suffered 
engine failures and wound up flying on three engines instead of four.  
Loud reports of every equipment failure imaginable were heard, including 
ejection seat problems.  It seems that the Blackjack remains a premature 
aircraft; that could explain the retention of cruise-missile-armed 
Tu-95MSs in the strategic aviation inventory until they get all the bugs 
ironed out.

An interesting aside is that the newly appointed Soviet Air Force 
commander (a strategic aviation pilot himself with over 4,000 hours in 
Backfires and Blackjacks) tried to keelhaul Tupolev officials for 
delivering a defective product that hadn't been fully tested before 
being dumped into service.  This was during the heady days of late-stage 
glasnost; there was a big scandal, flight safety issues came to the fore, 
etc.  Not sure if it went any further, though -- the Tu-160s are still in 
the inventory.
From: mstucka@orion.it.luc.edu (Mike Stucka)

In theory, they had gotten most of the bugs worked out.  The Tu-160 
project was derived in very large  part from the Tu-144SST 
("Concordski"), which was known for its failures.  Like the tires -- lots 
of 'em, too -- lasted just a  few landings, or shall we mention the one 
that literally fell apart in air at the Paris Air Show?

If you want more info, look for "Soviet SST" by Howard Moon.

Not sure what happened with it, but a year or two ago the Ukraine was 
looking to sell theirs.

From: agretch@opie.bgsu.edu (Alexei B. Gretchikhine)

AFAK, There are only handfull of these beasts in the inventory of DA. Actually,
most of them were kept by Ukraine (16 a/c) and Russian AF has only 2. Ukraine
wants to pay off its oil debt by selling these (plus few more Tu-95MS) to
Russia. I am not sure if deal took place yet.

I agree that Tu-160 is known for its bugs, but guy's look at B-1. People in
this group were b*ching how unreliable Lancer is.

Crews hate to fly Tu-160. Maintenance people do to.
I am surprised though, that Tu-160 has a smaller payload than B-1. It is much
larger a/c afterall.

I don't think there is much in common between Tu-144 and Tu-160. They even use
different engines. I would not deny that flight hours accumulated on SST helped
in design of BlackJack, but this is probably about it.

BTW, Tu-144 was FIRST SST to fly. It fell apart on
the Paris AirShow because it departed its flight envelop (due to copilot
leaning on the yoke). It is fine a/c.
From: m_cram@trofs.enet.dec.com (Marshall Cram)
>mstucka@orion.it.luc.edu (Mike Stucka) wrote:
> 
>The TU-144 used turbojet engines of questionable quality and vintage... the
> Blackjack uses Soloviev turbofans, I believe. I assume since it's top
>speed is purported to be slightly higher than that of the B1-B that these
>are pretty powerful, however their maintenance record I do not know,
>although the bomber is purported to be a fine plane in most respects.

The Tu-160 Blackjack uses the new, three-shaft Kuznetsov NK-321.
It's an afterburning turbofan of 55,115 lbs. thrust.

The Tu-144 used the Kuznetsov NK-144 (38,580 lbs.), NK-144A (44,092 lb.), 
and the NK-144B (48,500 lb.) progressively.  Two-shaft afterburning turbofans.

The Tu-22M Backfire engine, the Kuznetsov NK-22, has the same thrust
and fan diameter
as the NK-144B Tu-144 engine, indicating they were the same essential design,
and that they must have got a fairly reliable, if thirsty, engine in the end.

Some vital numbers:

Power Plant:
4xSamara/Trud NK-321 turbofans; 50,580 lb trust each with reheat.
Dimentions:
Span 182 ft 9 in spread, 116 ft 9 in swept; lenght 177 ft 6 in; height 43 ft
Weights:
Empty 242,500 lb, gross 606,260 lb.
Performance:
Max speed @ high alt. Mach 1.88, cruse 497 mph, celling 60,000 ft, 
range 7,455 miles (provision for in-flight refueling)
Accomodation:
Crew of four in pairs on ejection seats.
Armament:
Internally up to 88,185 lb of bombs, SRAMs, ALCMs. 
Each rotary launcher carries 12 Kh-15P (AS-16 "Kickback") SRAMs or
six Kh-55 (AS-15 "Kent") ALCMs.

For more info see:
Aviation Week and Space Technology Aug 15,1988, p. 16-18;
Aug 8,1988, p. 14-15; May 16,1988, p. 50-55; May 9,1988, p. 43-45;
Aug 24, 1992, p.65.
Time 1988, Aug 15, p. 21.