Inclusion Criteria :

          1. Histology:

               -  Dose escalation part: histologically confirmed non MSI-H or deficient-MMR
                  colorectal cancer, or GIST, or esophageal or gastric carcinoma or hepatobiliary
                  cancers,

               -  Phase II trials : histologically confirmed

                    -  non MSI-H or deficient-MMR colorectal cancer (cohort A),

                    -  or GIST (cohort B). As recommended diagnosis by INCa, patients with GIST
                       must have histologically confirmed by central review, except if it has been
                       already confirmed by the RRePS Network

                    -  or esophageal or gastric carcinoma (cohort C),

                    -  or hepatobiliary cancers (cohort D),

                    -  or soft-tissue sarcoma (STS) (cohort E). As recommended diagnosis by INCa,
                       patients with STS must have histologically confirmed by central review,
                       except if it has been already confirmed by the RRePS Network

                    -  or radioiodine-refractory differentiated thyroid cancer [RR-DTC] (cohort F),

                    -  or neuroendocrine gastroenteropancreatic tumors grade 2 and 3

                    -  or Non-small cell lung cancer (cohort H)

                    -  or Solid tumors including soft-tissue sarcoma with immune signature (cohort
                       I), i.e. presence of tertiary lymphoid structures on tumor sample as
                       determined by central review.

          2. Advanced non resectable / metastatic disease,

          3. Patients for which either there is no further established therapy that is known to
             provide clinical benefit, OR (for patients to be treated with 160 mg regorafenib)
             regorafenib monotherapy is an approved or established therapeutic option,

          4. Age ≥ 18 years,

          5. ECOG, Performance status ≤ 1,

          6. Measurable disease according to RECIST,

          7. Life expectancy > 3 months,

          8. Except for cohorts F and H, ≥ 1 previous line (s) of systemic therapy,

          9. Adequate hematological, renal, metabolic and hepatic functions:

               1. Hemoglobin ≥ 9 g/dl (patients may have received prior red blood cell [RBC]
                  transfusion, if clinically indicated); absolute neutrophil count (ANC) ≥ 1.5 x
                  109/l and platelet count ≥ 100 x 109/l, lymphocytes ≥ 1000/mm3.

               2. Alkaline phosphatase (AP), alanine aminotransferase (ALT) and aspartate
                  aminotransferase (ASP) ≤ 2.5 x upper limit of normality (ULN) (≤ 5 in case of
                  extensive skeletal involvement for AP exclusively and ≤ 5 x ULN in case of liver
                  metastasis for AST and ALT).

               3. Total bilirubin ≤ 1.5 x ULN.

               4. Albumin ≥ 25g/l.

               5. Calculated creatinine clearance (CrCl) ≥ 30 ml/min (according to Cockroft and
                  Gault formula).

               6. Creatine phosphokinase (CPK) ≤ 2.5 x ULN.

               7. INR < 1.5 x ULN

               8. aPTT ≤ 1.5 X ULN

               9. Lipase ≤ 1.5 X ULN.

              10. Cohort specific criteria: Patients with hepatocellular carcinoma must have a
                  correct hepatocellular function, id est Child-Pugh A.

         10. No prior or concurrent malignant disease diagnosed or treated in the last 2 years
             except for adequately treated in situ carcinoma of the cervix, basal or squamous skin
             cell carcinoma, or in situ transitional bladder cell carcinoma,

         11. At least three weeks since last chemotherapy, immunotherapy or any other
             pharmacological treatment and/or radiotherapy,

         12. Recovery to grade ≤ 1 from any adverse event (AE) derived from previous treatment,
             excluding alopecia of any grade and non-painful peripheral neuropathy grade ≤ 2
             (according to the National Cancer Institute Common Terminology Criteria for Adverse
             Event (NCI-CTCAE, version 5.0)),

         13. Women of childbearing potential must have a negative serum pregnancy test within 72
             hours prior to receiving the first dose of study medication,

         14. Both women and men must agree to use an highly effective method of contraception
             throughout the treatment period and for eight weeks after discontinuation of
             treatment. Acceptable methods for contraception are described in protocol section
             7.4.1,

         15. Voluntary signed and dated written informed consents prior to any specific study
             procedure,

         16. Patients with a social security in compliance with the French law.

         17. Documented disease progression (as per RECIST v1.1) before study entry. For patients
             of cohort E (STS) and cohort I (Solid-tumors - TLS+): this progression will be
             confirmed by central review on the basis of two CT scan or MRI obtained not less than
             6 months in the period of 12 months prior to inclusion. For patients of cohort F
             (RR-DTC): this progression will be confirmed by central review on the basis of two CT
             scan or MRI obtained at less than 12 months prior to inclusion.

         18. For patients in cohort H: subjects with histologically or cytologically confirmed
             diagnosis of non-squamous NSCLC. Documents disease progression based on radiographic
             imaging, during or after a maximum of 2 lines of systemic treatment for
             locally/regionally advanced recurrent, Stage IIIb/IV or metastatic disease. Two
             components of treatments must have been received in the same line or as separate lines
             of therapy: a maximum of 1 line of platinum-containing chemotherapy regimen, and a
             maximum of 1 line of PD(L)1 mAb containing regimen. No EGFR, ALK, ROS1 positive tumor
             mutations. Subjects with known BRAF molecular alterations must have had disease
             progression after receiving the locally available SoC treatment for the molecular
             alteration

        Exclusion Criteria:

          1. Previous treatment with Avelumab or Regorafenib,

          2. Has received prior therapy with an anti-PD-1, anti-PD-L1, anti-PD-L2, anti- CD137, or
             anti-Cytotoxic T-lymphocyte-associated antigen-4 (CTLA-4) antibody (including
             ipilimumab or any other antibody or drug specifically targeting T-cell costimulation
             or checkpoint pathways), except for cohort H

          3. Evidence of progressive or symptomatic or newly diagnosed central nervous system (CNS)
             or leptomeningeal metastases,

          4. Men or women of childbearing potential who are not using an effective method of
             contraception as previously described;

          5. Participation to a study involving a medical or therapeutic intervention in the last
             30 days,

          6. Previous enrolment in the present study,

          7. Patient unable to follow and comply with the study procedures because of any
             geographical, familial, social or psychological reasons,

          8. Known hypersensitivity to any involved study drug or of its formulation components,

          9. Active autoimmune disease that might deteriorate when receiving an immunostimulatory
             agent :

               1. Subjects with diabetes type I, vitiligo, psoriasis, hypo- or hyperthyroid disease
                  not requiring immunosuppressive treatment are eligible

               2. Subjects requiring hormone replacement with corticosteroids are eligible if the
                  steroids are administered only for the purpose of hormonal replacement and at
                  doses ≤ 10 mg or 10 mg equivalent prednisone per day

               3. Administration of steroids through a route known to result in a minimal systemic
                  exposure (topical, intranasal, intro-ocular, or inhalation) are acceptable

         10. Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any
             other form of immunosuppressive therapy within 7 days prior to the first dose of trial
             treatment,

         11. History of idiopathic pulmonary fibrosis (including pneumonitis), drug-induced
             pneumonitis, organizing pneumonia, or evidence of active pneumonitis on screening
             chest CT scan or interstitial lung disease with ongoing signs and symptoms at
             inclusion. History of radiation pneumonitis in the radiation field (fibrosis) is
             permitted,

         12. Has known hepatitis B or hepatitis C, active and/or treated by antiviral therapy

         13. Has a known history of Human Immunodeficiency Virus (HIV) (HIV1/2 antibodies) or known
             acquired immunodeficiency syndrome (AIDS),

         14. Spot urine must not show 1+ or more protein in urine or the patient will require a
             repeat urine analysis,

         15. Major surgical procedure or significant traumatic injury within 28 days before start
             of study medication,

         16. Non-healing wound, non-healing ulcer, or non-healing bone fracture requiring
             orthopedic treatment,

         17. Patients with evidence or history of any bleeding diathesis, irrespective of severity,

         18. Any hemorrhage or bleeding event ≥ CTCAE Grade 3 within 4 weeks prior to the start of
             study medication,

         19. Arterial or venous thrombotic or embolic events such as cerebrovascular accident
             (including transient ischemic attacks), deep vein thrombosis or pulmonary embolism
             within 6 months before the start of study medication (except for adequately treated
             catheter-related venous thrombosis occurring more than one month before the start of
             study medication),

         20. Ongoing infection > Grade 2 as per NCI CTCAE v5.0,

         21. Uncontrolled hypertension (Systolic blood pressure > 140 mmHg or diastolic pressure >
             90 mmHg) despite optimal medical management,

         22. Congestive heart failure ≥ New York Heart Association (NHYA) class 2,

         23. Unstable angina (angina symptoms at rest), new-onset angina (begun within the last 3
             months),

         24. Myocardial infarction less than 6 months bedfore start of study drug,

         25. Uncontrolled cardiac arrhythmias,

         26. Pregnant or breast-feeding patients,

         27. Individuals deprived of liberty or placed under legal guardianship,

         28. Prior organ transplantation, including allogeneic stem-cell transplantation,

         29. Known alcohol or drug abuse,

         30. Vaccination within 4 weeks of the first dose of Avelumab and while on trial is
             prohibited except for administration of inactivated vaccines,

         31. Patients with any condition that impairs their ability to swallow and retain tablets,

         32. Other severe acute or chronic medical conditions including immune colitis,
             inflammatory bowel disease, immune pneumonitis, pulmonary fibrosis or psychiatric
             conditions including recent (within the past year) or active suicidal ideation or
             behavior; or laboratory abnormalities that may increase the risk associated with study
             participation or study treatment administration or may interfere with the
             interpretation of study results and, in the judgment of the investigator, would make
             the patient inappropriate for entry into this study,

         33. Patient with oral anticoagulation therapy,

         34. Suspected or known intraabdominal fistula.

         35. For cohort H: received more than 2 prior lines of therapy for NSCLC, including
             subjects with BRAF molecular alteration and subjects with knwon EGFR/ALK/ROS1
             molecular alterations are excluded