When a country adopts the Euro, what happens to its debt? When a country, say Greece, adopted the Euro as its currency, what happened to the debt that was based on the Drachma? Was it converted into Euros?
Yes, upon the introduction of the euro on January 1, 1999, all debt (indeed, all nominal contracts) in participating countries was converted from national currency to euros at a legally defined conversion rate. See this press release from December 31, 1998, which states:


  In accordance with Article 109l (4) of the Treaty establishing the
  European Community, the irrevocable conversion rates for the euro were
  today adopted by the EU Council, upon a proposal from the Commission
  of the European Communities and after consultation of the European
  Central Bank (ECB) for effect at 0.00 on 1 January 1999 (local time).
  In compliance with the legal framework for the use of the euro, the
  irrevocable conversion rate for the euro for each participating
  currency is the only rate to be used for conversion either way between
  the euro and the national currency unit and also for conversions
  between national currency units.
  
  The euro conversion rates are the following...


For the particular case of Greece, which joined the euro on January 1, 2001 instead, the conversion rate with the drachma was apparently set at 1 euro = 340.750 drachma by this regulation in June 2000.