An apparatus for targeting a desired target site on a body tissue comprising a wall of a first body cavity of a patient, the apparatus comprising: a target catheter having a longitudinally extending target catheter lumen surrounded by a tubular target catheter wall having an outer surface; a framing member having a collapsed condition in which the framing member is adapted for insertion into the first body cavity and an expanded condition in which the framing member is adapted for placement within the first body cavity, the framing member, when deployed into the expanded condition, having: a framing member body which includes a three-dimensionally bulbous portion defining a maximum body footprint in a lateral dimension and having longitudinally spaced proximal and distal body ends (2874, 2876) which are both longitudinally spaced from the maximum body footprint, a diameter of the bulbous portion being significantly smaller at the proximal and distal body ends (2874, 2876) than at the maximum body footprint, the proximal body end being attached to the outer surface of the target catheter wall; at least one target point carried by the framing member and adapted for placement adjacent the desired target site; and at least one target pathway attached to at least one target point, at least a portion of the target pathway extending through the target catheter lumen, and the target pathway being substantially spaced apart from the framing member body; wherein the framing member is comprised of a plurality of framing member beams, with at least three framing member beam arrays of framing member beams arranged in non-overlapping longitudinal sequence to form at least a portion of the framing member; wherein the framing member beams that make up the framing member substantially each have a width-to-thickness fractional ratio in the range of 0.6-3.0; and wherein at least one framing member beam of a chosen framing member beam array (4304, 4304C, 4304D, 4304P) includes at least one longitudinal slit along a portion of a longitudinal length of the at least one framing member beam. a target catheter having a longitudinally extending target catheter lumen surrounded by a tubular target catheter wall having an outer surface; a framing member having a collapsed condition in which the framing member is adapted for insertion into the first body cavity and an expanded condition in which the framing member is adapted for placement within the first body cavity, the framing member, when deployed into the expanded condition, having: a framing member body which includes a three-dimensionally bulbous portion defining a maximum body footprint in a lateral dimension and having longitudinally spaced proximal and distal body ends (2874, 2876) which are both longitudinally spaced from the maximum body footprint, a diameter of the bulbous portion being significantly smaller at the proximal and distal body ends (2874, 2876) than at the maximum body footprint, the proximal body end being attached to the outer surface of the target catheter wall; at least one target point carried by the framing member and adapted for placement adjacent the desired target site; and at least one target pathway attached to at least one target point, at least a portion of the target pathway extending through the target catheter lumen, and the target pathway being substantially spaced apart from the framing member body; wherein the framing member is comprised of a plurality of framing member beams, with at least three framing member beam arrays of framing member beams arranged in non-overlapping longitudinal sequence to form at least a portion of the framing member; wherein the framing member beams that make up the framing member substantially each have a width-to-thickness fractional ratio in the range of 0.6-3.0; and wherein at least one framing member beam of a chosen framing member beam array (4304, 4304C, 4304D, 4304P) includes at least one longitudinal slit along a portion of a longitudinal length of the at least one framing member beam.