A modular valve replacement system implantable into a heart using a catheter-based implantation system, the modular valve replacement system comprising: a fixation device (810; 910) including: an outer structure having a tissue engagement portion (830; 930), an upstream portion, and a downstream portion, wherein the tissue engagement portion is configured to engage native heart tissue when the fixation device is deployed within the heart to independently secure the fixation device to the native heart tissue, and a temporary valve assembly including (a) an inner structure having a generally toroidal shape when the fixation device is in the deployed configuration, wherein the inner structure includes a chamber configured to receive blood when the fixation device is implanted at or adjacent the native valve annulus, and (b) a temporary prosthetic valve (848; 948) attached to the inner structure; and a permanent valve assembly (850; 950) having a valve support (860; 960) and a prosthetic valve (870; 970) attached to the valve support, wherein the permanent valve assembly is separate from the fixation device in a low-profile state for delivery via a catheter and configured to be connected to the fixation device in vivo after the fixation device has been deployed within the heart, and the valve support is spaced inwardly apart from the tissue engagement portion when the permanent valve assembly is attached to the fixation device in vivo. a fixation device (810; 910) including: an outer structure having a tissue engagement portion (830; 930), an upstream portion, and a downstream portion, wherein the tissue engagement portion is configured to engage native heart tissue when the fixation device is deployed within the heart to independently secure the fixation device to the native heart tissue, and a temporary valve assembly including (a) an inner structure having a generally toroidal shape when the fixation device is in the deployed configuration, wherein the inner structure includes a chamber configured to receive blood when the fixation device is implanted at or adjacent the native valve annulus, and (b) a temporary prosthetic valve (848; 948) attached to the inner structure; and an outer structure having a tissue engagement portion (830; 930), an upstream portion, and a downstream portion, wherein the tissue engagement portion is configured to engage native heart tissue when the fixation device is deployed within the heart to independently secure the fixation device to the native heart tissue, and a temporary valve assembly including (a) an inner structure having a generally toroidal shape when the fixation device is in the deployed configuration, wherein the inner structure includes a chamber configured to receive blood when the fixation device is implanted at or adjacent the native valve annulus, and (b) a temporary prosthetic valve (848; 948) attached to the inner structure; and a permanent valve assembly (850; 950) having a valve support (860; 960) and a prosthetic valve (870; 970) attached to the valve support, wherein the permanent valve assembly is separate from the fixation device in a low-profile state for delivery via a catheter and configured to be connected to the fixation device in vivo after the fixation device has been deployed within the heart, and the valve support is spaced inwardly apart from the tissue engagement portion when the permanent valve assembly is attached to the fixation device in vivo.