NUVEEN HIGH YIELD INCOME FUND

NUVEEN FLOATING RATE INCOME FUND

SUPPLEMENT DATED DECEMBER 2, 2020

TO THE PROSPECTUS AND SUMMARY PROSPECTUSES DATED JANUARY 31, 2020

In connection with the proposed merger of Symphony Asset Management LLC (“Symphony”) with and into Nuveen Asset Management, LLC (“NAM”) (the “Reorganization”), which is currently scheduled to close on December 31, 2020, the Board of Trustees of Nuveen Investment Trust III has approved amended and restated sub-advisory agreements, effective as of the closing date of the Reorganization, between Nuveen Fund Advisors, LLC (“NFAL”), as investment adviser of Nuveen High Yield Income Fund and Nuveen Floating Rate Income Fund, and NAM, pursuant to which NAM will replace Symphony as the Funds’ sub-adviser. NAM and Symphony are both affiliates of NFAL and are subsidiaries of Nuveen, LLC.

Scott Caraher and Jean Lin will continue to serve as portfolio managers of Nuveen High Yield Income Fund and Mr. Caraher and Kevin Lorenz will continue to serve as portfolio managers of Nuveen Floating Rate Income Fund.

PLEASE KEEP THIS WITH YOUR

FUND’S PROSPECTUS AND/OR SUMMARY PROSPECTUS

FOR FUTURE REFERENCE

MGN-SCFRSP-1220P

NUVEEN HIGH YIELD INCOME FUND

NUVEEN FLOATING RATE INCOME FUND

SUPPLEMENT DATED DECEMBER 2, 2020

TO THE STATEMENT OF ADDITIONAL INFORMATION

DATED JANUARY 31, 2020

In connection with the proposed merger of Symphony Asset Management LLC (“Symphony”) with and into Nuveen Asset Management, LLC (“NAM”) (the “Reorganization”), which is currently scheduled to close on December 31, 2020, the Board of Trustees of Nuveen Investment Trust III has approved amended and restated sub-advisory agreements, effective as of the closing date of the Reorganization, between Nuveen Fund Advisors, LLC (“NFAL”), as investment adviser of Nuveen High Yield Income Fund and Nuveen Floating Rate Income Fund, and NAM, pursuant to which NAM will replace Symphony as the Funds’ sub-adviser. NAM and Symphony are both affiliates of NFAL and are subsidiaries of Nuveen, LLC.

Scott Caraher and Jean Lin will continue to serve as portfolio managers of Nuveen High Yield Income Fund and Mr. Caraher and Kevin Lorenz will continue to serve as portfolio managers of Nuveen Floating Rate Income Fund.

PLEASE KEEP THIS WITH YOUR

FUND’S STATEMENT OF ADDITIONAL INFORMATION

FOR FUTURE REFERENCE

MGN-SCFRSAI-1220P