Bobby Brown made his debut as a goalkeeper for Queen's Park in 1939. He went on to establish himself as the first-choice goalkeeper for the next two seasons, but like many others at this time, his appearances were interrupted by the Second World War, where he served in the Navy as a physical training instructor. During this time, he played as a guest in wartime leagues for Portsmouth, Chester, Chelsea and Plymouth Argyle.
Brown played in several war time internationals, and in his last season at Hampden in season 1945–46 he shared the Queen's Park goalkeeper's jersey with another future Scottish international, Ronnie Simpson. His form was rewarded with a full international call up, and in January 1946 he made his Scotland debut in a friendly against Belgium. He remains the last amateur player to earn a full international cap for Scotland, and is also the last player to earn a full Scotland cap whilst still a signed player at Hampden.
At the end of the 1945–46 season, he left Queen's Park for Rangers, where he played for ten years. During his time at Rangers, he won five League championships and four Scottish Cups. He also won a further four caps for Scotland before moving to Falkirk in 1956. Brown appeared eight times for the Scottish League XI between 1949 and 1952. He played for Falkirk for two years, after which he retired from playing.
Brown then became manager of St Johnstone. He became the first full-time Scotland manager in 1967, and his first international match as manager was the famous 3–2 victory over world champions England at Wembley Stadium. This game also saw Brown give his goalkeeping understudy from his Hampden days, Ronnie Simpson, his international debut. Bobby continued as Scotland manager until 1971, but often found his Scotland squads depleted by club demands and results suffered accordingly. After finishing with Scotland in 1971, he turned to his business interests outside football, although he remained involved in the game, scouting for Plymouth Argyle for several years. He is now retired and lives in Helensburgh.