Moishe Leopoldowicz Łowcki was brought up in Kiev, and then studied at its university. He took part in Third Russian National Tournament (1903) in Kiev taking 6-7th place (together with Evgeny Znosko-Borowski).

Łowcki studied in Germany and played in Germany events. In 1903, he was 4th of eight in the Dresden Saxonian Chess Tournament for amateurs; a year later at the chess festival in Coburg he took 2nd place. He performed successfully in subsequent tournaments: in 1910 he won the qualifying rounds in Hamburg, and in the main event he came in third in 14th German Chess Federation Congress (Hauptturnier C). In the same year at the Augustea Winter Tournament in Leipzig he divided 1-3 places, finally in 1911 he triumphed in an amateur tournament in Cologne. These results brought Łowcki gave him access to prestigious tournaments for the European leaders. In 1911 in San Remo, Łowcki took second place, also he was runner-up. In 1912, he took the 5th place at the gambit tournament in Abbazia (1912), won by Rudolf Spielmann. Later that year he was 12th in Bad Pistyan (1912) and 16th in the 18th DSB Kongress (1912). The All-Russian Championship (1914), St. Petersburg, Łowcki was 4th behind Alexander Alekhibe, Aron Nimzowitsch and Alexander Flamberg.

Łowcki lost two matches to Leonhardt in 1910 and 1913

In 1913 Łowcki settled permanently in Warsaw and was an active participant in the chess cafe life and 1910 -1915 were his peak years. Łowcki was the runner-up in the Warsaw City Championships in 1916 and 1917. In 1922 he lost a match to Flamberg, but after the First World War, his appearance in tournaments declined and were confined to Polish events.

He was arrested by the Gestapo in January 1940 and killed in a mass execution in Kampinos Forest.