A system for forming an anastomosis between two lumens (520, 30) or two locations in a lumen of a patient, the system comprising: a first magnetic member configured to be located adjacent a first luminal tissue region at a first location; and a second magnetic member configured to be located adjacent a second luminal tissue region at a second location, wherein: the first magnetic member has a first engagement surface that faces a second engagement surface of the second magnetic member when the first magnetic member and the second magnetic member are magnetically coupled to one another with the first and second luminal tissue regions interposed therebetween, the first engagement surface has a first average radial cross-sectional profile, the second engagement surface has a second average radial cross-sectional profile, a linear distance between the first average radial cross-sectional profile and the second average radial cross-sectional profile increases with increasing radial distance from an innermost portion of the first and second magnetic members, and characterized in that the first engagement surface and the second engagement surface are configured, when the first magnetic member and the second magnetic member are magnetically coupled together with the first and second luminal tissue regions interposed therebetween, to exert a pressure of between 20 kPa and 80 kPa on nominally annular first portions of the first and second luminal tissue regions located along interior edges (314, 324) of the first and second magnetic members and a pressure of 0 kPa to 15 kPa along nominally annular second portions of the first and second luminal tissue regions located along exterior edges (312, 322) of the first and second magnetic members. a first magnetic member configured to be located adjacent a first luminal tissue region at a first location; and a second magnetic member configured to be located adjacent a second luminal tissue region at a second location, wherein: the first magnetic member has a first engagement surface that faces a second engagement surface of the second magnetic member when the first magnetic member and the second magnetic member are magnetically coupled to one another with the first and second luminal tissue regions interposed therebetween, the first engagement surface has a first average radial cross-sectional profile, the second engagement surface has a second average radial cross-sectional profile, a linear distance between the first average radial cross-sectional profile and the second average radial cross-sectional profile increases with increasing radial distance from an innermost portion of the first and second magnetic members, and characterized in that the first engagement surface and the second engagement surface are configured, when the first magnetic member and the second magnetic member are magnetically coupled together with the first and second luminal tissue regions interposed therebetween, to exert a pressure of between 20 kPa and 80 kPa on nominally annular first portions of the first and second luminal tissue regions located along interior edges (314, 324) of the first and second magnetic members and a pressure of 0 kPa to 15 kPa along nominally annular second portions of the first and second luminal tissue regions located along exterior edges (312, 322) of the first and second magnetic members. the first magnetic member has a first engagement surface that faces a second engagement surface of the second magnetic member when the first magnetic member and the second magnetic member are magnetically coupled to one another with the first and second luminal tissue regions interposed therebetween, the first engagement surface has a first average radial cross-sectional profile, the second engagement surface has a second average radial cross-sectional profile, a linear distance between the first average radial cross-sectional profile and the second average radial cross-sectional profile increases with increasing radial distance from an innermost portion of the first and second magnetic members, and characterized in that the first engagement surface and the second engagement surface are configured, when the first magnetic member and the second magnetic member are magnetically coupled together with the first and second luminal tissue regions interposed therebetween, to exert a pressure of between 20 kPa and 80 kPa on nominally annular first portions of the first and second luminal tissue regions located along interior edges (314, 324) of the first and second magnetic members and a pressure of 0 kPa to 15 kPa along nominally annular second portions of the first and second luminal tissue regions located along exterior edges (312, 322) of the first and second magnetic members.