The Chazon Ish zt'l lit the Chanukah lights on weekdays twenty minutes after sunset. On Motzei Shabbos, however, he lit them only after ma'ariv (which he davened approximately three quarters of an hour after sunset) and making havdoloh. [The report in Tohoras Habayis (part 2, p. 273) that the Chazon Ish lit Chanukah lights twenty minutes after sunset even on Motzei Shabbos--from which the author deduces that melochos may be done this soon after sunset when there is a pressing necessity--is erroneous].

The source of the minhag of the Chazon Ish of when to light Chanukah lights on weekdays is found in the Bei'ur HaGra in Yoreh Dei'ah, Hilchos Milah chap. 266:17, who explicitly mentions Chanukah lights!

The Chazon Ish based his reasoning on the fact that there must be some darkness for Chanukah lighting, since "what good is a candle during the day?" In central Eretz Yisroel we light 20-21 minutes after sunset (depending when Chanukah falls according to the solar calendar). The sun is at that time 4.8 degrees below the horizon.

Using the sun's position below the horizon as our criterion, these are the times to light Chanukah lights in selected cities worldwide according to the Chazon Ish: Zurich 27-28 minutes after sunset; Vienna 28-29 minutes; Sydney 22 minutes; Strasbourg 28 minutes; Stockholm 43-44 minutes; Rome 24-25 minutes; Paris 28-29 minutes; New York 23-24 minutes; Gateshead 35 minutes; Melbourne 23-24 minutes; Manchester 33 minutes; Los Angeles 21-22 minutes; London 31 minutes; Johannesburg 20 minutes; Cleveland 24 minutes; Chicago 24-25 minutes; Buenos Aires 22-23 minutes; Amsterdam 32 minutes.

The custom in Yerushalayim, based on the traditions of the talmidei Hagra, is to light at sunset.