Apparatus comprising a mitral valve prosthesis for implantation at a native mitral valve complex of a subject, the prosthesis comprising: an inner support structure having a downstream section and an upstream section , the upstream section having a cross-sectional area greater than the downstream section, the inner support structure being configured to be positioned at least partially on an atrial side of the native valve complex, and to prevent the prosthesis from being dislodged into a left ventricle by applying an axial force directed toward the left ventricle; a prosthetic valve having prosthetic valve leaflets coupled to the inner support structure; and an outer support structure having two engagement arms , each engagement arm being concave in a downstream direction, the downstream ends of each of the engagement arms being coupled to the inner support structure, wherein a longitudinal distance (D1) from the downstream end to an upstream end of each of the engagement arms is less than 18 mm, wherein the downstream ends of the engagement arms are coupled to the inner support structure within 3 mm of a downstream end of the inner support structure, and wherein the prosthetic valve leaflets are coupled to the inner support structure such that downstream ends of the prosthetic valve leaflets are within 3 mm of the downstream ends of said engagement arms, the prosthesis being configured such that, upon implantation thereof with the native valve leaflets of the native mitral valve complex disposed between the engagement arms and the inner support structure: downstream ends of native valve leaflets of the native mitral valve complex, the downstream ends of the engagement arms, and downstream ends of the prosthetic leaflets, are disposed at a longitudinal distance from one another of less than 3 mm, the longitudinal distances being measured in a direction of the longitudinal axis of the prosthesis. an inner support structure having a downstream section and an upstream section , the upstream section having a cross-sectional area greater than the downstream section, the inner support structure being configured to be positioned at least partially on an atrial side of the native valve complex, and to prevent the prosthesis from being dislodged into a left ventricle by applying an axial force directed toward the left ventricle; a prosthetic valve having prosthetic valve leaflets coupled to the inner support structure; and an outer support structure having two engagement arms , each engagement arm being concave in a downstream direction, the downstream ends of each of the engagement arms being coupled to the inner support structure, wherein a longitudinal distance (D1) from the downstream end to an upstream end of each of the engagement arms is less than 18 mm, wherein the downstream ends of the engagement arms are coupled to the inner support structure within 3 mm of a downstream end of the inner support structure, and wherein the prosthetic valve leaflets are coupled to the inner support structure such that downstream ends of the prosthetic valve leaflets are within 3 mm of the downstream ends of said engagement arms, the prosthesis being configured such that, upon implantation thereof with the native valve leaflets of the native mitral valve complex disposed between the engagement arms and the inner support structure: downstream ends of native valve leaflets of the native mitral valve complex, the downstream ends of the engagement arms, and downstream ends of the prosthetic leaflets, downstream ends of native valve leaflets of the native mitral valve complex, the downstream ends of the engagement arms, and downstream ends of the prosthetic leaflets, are disposed at a longitudinal distance from one another of less than 3 mm, the longitudinal distances being measured in a direction of the longitudinal axis of the prosthesis. <EOS>