FORM 4 Check this box if no longer subject to Section 16. Form 4 or Form 5 obligations may continue. See Instruction 1(b).

UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 STATEMENT OF CHANGES IN BENEFICIAL OWNERSHIP OF SECURITIES Filed pursuant to Section 16(a) of the Securities Exchange Act of 1934, Section 17(a) of the Public Utility Holding Company Act of 1935 or Section 30(h) of the Investment Company Act of 1940

OMB APPROVAL OMB Number: 3235-0287 Expires: January 31, 2005 Estimated average burden hours per response... 0.5

(Print or Type Responses)

1. Name and Address of Reporting Person * GREENWOOD DAVID   2. Issuer Name and Ticker or Trading Symbol GERON CORP [GERN] 5. Relationship of Reporting Person(s) to Issuer (Check all applicable) _____ Director _____ 10% Owner __X__ Officer (give title below) _____ Other (specify below) EVP and CFO (Last) (First) (Middle) C/O GERON CORPORATION, 230 CONSTITUTION DRIVE 3. Date of Earliest Transaction (Month/Day/Year) 05/23/2008 (Street) MENLO PARK, CA 94025 4. If Amendment, Date Original Filed(Month/Day/Year) 6. Individual or Joint/Group Filing(Check Applicable Line) _X_ Form filed by One Reporting Person ___ Form filed by More than One Reporting Person (City) (State) (Zip) Table I - Non-Derivative Securities Acquired, Disposed of, or Beneficially Owned

1.Title of Security (Instr. 3) 2. Transaction Date (Month/Day/Year) 2A. Deemed Execution Date, if any (Month/Day/Year) 3. Transaction Code (Instr. 8) 4. Securities Acquired (A) or Disposed of (D) (Instr. 3, 4 and 5) 5. Amount of Securities Beneficially Owned Following Reported Transaction(s) (Instr. 3 and 4) 6. Ownership Form: Direct (D) or Indirect (I) (Instr. 4) 7. Nature of Indirect Beneficial Ownership (Instr. 4) Code V Amount (A) or (D) Price Common Stock 05/23/2008   F(1)   18,991 (1) D $ 3.92 (1) 152,792 D   Common Stock 05/28/2008   A   37,500 (2) A $ 0 (2) 190,292 D

Reminder: Report on a separate line for each class of securities beneficially owned directly or indirectly.
	Persons who respond to the collection of information contained in this form are not required to respond unless the form displays a currently valid OMB control number.		SEC 1474 (9-02)

Table II - Derivative Securities Acquired, Disposed of, or Beneficially Owned
(e.g., puts, calls, warrants, options, convertible securities)

1. Title of Derivative Security (Instr. 3)

2. Conversion or Exercise Price of Derivative Security

3. Transaction Date (Month/Day/Year)

3A. Deemed Execution Date, if any (Month/Day/Year)

4. Transaction Code (Instr. 8)

5. Number of Derivative Securities Acquired (A) or Disposed of (D) (Instr. 3, 4, and 5)

6. Date Exercisable and Expiration Date (Month/Day/Year)

7. Title and Amount of Underlying Securities (Instr. 3 and 4)

8. Price of Derivative Security (Instr. 5)

9. Number of Derivative Securities Beneficially Owned Following Reported Transaction(s) (Instr. 4)

10. Ownership Form of Derivative Security: Direct (D) or Indirect (I) (Instr. 4)

11. Nature of Indirect Beneficial Ownership (Instr. 4)

Code

V

(A)

(D)

Date Exercisable

Expiration Date

Title

Amount or Number of Shares

Stock Option (right to buy)

$ 3.97

05/28/2008

A

75,000

(3)

05/28/2018

Common Stock

75,000

$ 3.97

1,176,341

D

Stock Option (right to buy)

$ 3.97

05/28/2008

A

181,341

(4)

05/28/2018

Common Stock

181,341

$ 3.97

1,357,682

D

Stock Option (right to buy)

$ 3.97

05/28/2008

A

40,000

(5)

05/28/2018

Common Stock

40,000

$ 3.97

1,397,682

D

Reporting Owners

Reporting Owner Name / Address	Relationships
	Director	10% Owner	Officer	Other
GREENWOOD DAVID C/O GERON CORPORATION 230 CONSTITUTION DRIVE MENLO PARK, CA 94025			EVP and CFO

Signatures

/s/ Olivia Bloom for David L. Greenwood		05/28/2008
**Signature of Reporting Person		Date

Explanation of Responses:

*	If the form is filed by more than one reporting person, see Instruction 4(b)(v).
**	Intentional misstatements or omissions of facts constitute Federal Criminal Violations. See 18 U.S.C. 1001 and 15 U.S.C. 78ff(a).
(1)	Shares equivalent to the amounts required to satisfy payroll tax withholdings are withheld from distribution to recipient upon vesting of stock award, reducing the vested amount from 53,125 to 34,134 shares. There is no sale of shares.
(2)	The shares of restricted stock will remain subject to a repurchase right of Geron over four years, during such time the repurchase right will lapse with respect to 25% of such shares on each anniversary date of the date of grant. The lapse of Geron's repurchase right is conditioned on the employee continuing to provide services to Geron through each vesting date.
(3)	This option is exercisable in a series of 48 equal monthly installments commencing May 28 2008 provided the optionee continues to provide services to the Company.
(4)	This option is exercisable in a series of 24 equal monthly installments commencing September 18, 2008, provided the optionee continues to provide services to the Company.
(5)	This option is exercisable in a series of 24 equal monthly installments commencing September 4, 2008, provided the optionee continues to provide services to the Company.

Note: File three copies of this Form, one of which must be manually signed. If space is insufficient, see Instruction 6 for procedure.

Potential persons who are to respond to the collection of information contained in this form are not required to respond unless the form displays a currently valid OMB number.