There's no single undisputed number — because "Europe" and "country" are both contested categories.
44 sovereign states with territory entirely or predominantly in geographic Europe (UN geoscheme). Add Vatican City (UN observer) = 45. This is the most common textbook answer.
46–47: Include Kosovo (recognized by ~100 UN members, not Serbia/UN) and/or Cyprus (geographically Asia, politically Europe).
49–51: Add transcontinental states with European territory/capital: Russia (77% land in Asia, but 75% population/European capital), Turkey (3% land in Europe, Istanbul straddles), Kazakhstan (small European foothold), Azerbaijan, Georgia, Armenia (Caucasus — Europe/Asia border debated).
52: Some counts include all Council of Europe members (46) + Belarus + Kazakhstan + Vatican + Kosovo = 50, plus micro-debates.
For general use: 44–45 (UN European sovereign states ± Vatican). For political/institutional contexts: specify the body (EU=27, CoE=46, etc.). For trivia: "It depends on where you draw the line."
The map is not the territory — and the count is not the continent.