There are ~167 city underground railway networks in the world. ~20 more are now planned or under construction (Dublin, Bologna, Odessa, Belgrade, Thessaloniki, Seattle, Kawasaki, Jakarta, Harbin, Tel Aviv, Algier). There were a few openings in 2010: Dubai in February, Adana in May, Shenyang in September, Mecca in November.
Not all metro systems have subway trains. Some have trams running underground, like in Antwerpen, but these also count. Not all the segments of metro lines are underground. Sometimes lines are short, but run all the way underground, like in Genova. Sometimes everything is above the ground, like in Jacksonville, but still it is called a metro.
By now I have travelled on 83 metros (listed below).
Prague, Budapest, St. Petersburg, Nürnberg, Helsinki, Copenhagen, London, München, Paris, Frankfurt/M, Singapore, Kuala Lumpur, Stockholm, Milan, Brussels, Antwerp, Rotterdam, Athens, Tokyo, Kyoto, Kobe, Osaka, Nagoya, Dusseldorf, Roma, Barcelona, Madrid, Marseilles, Lyon, Hamburg, Liverpool, ((Birmingham)), Moscow, Shanghai, Nanjing, Beijing, den Haag, Amsterdam, Lille, Koln, Berlin, New York, Philadelphia, Baltimore, Washington D.C., Atlanta, Jacksonville, Miami, Boston, Warsaw, Sofia, Genova, Toulouse, Hong Kong, Sydney, Wellington, Auckland, San Francisco, Glasgow, Newcastle, Shenzhen, Guangzhou, Rennes, Vancouver, Naples, Catania, Porto, Lisbon, Essen, Bochum, Dortmund, Hannover, Duisburg, Lausanne, Bonn, Bielefeld, Oslo, Istanbul, Kiev, Charleroi, Hiroshima, Yokohama, Wien