Inclusion Criteria:

          1. Written (signed and dated) informed consent and be capable of co-operating with
             treatment and follow-up.

          2. B-cell lymphoma or CLL proven by histology or flow cytometry, relapsed or refractory
             to conventional treatment, or for which no conventional therapy exists or is declined
             by the patient. Patients should have received at least one line of conventional
             previous therapy which must have included a rituximab based regimen.

          3. CD32b positive malignancy as demonstrated centrally by immunohistochemistry or flow
             cytometry prior to study entry. Available tissue or blood must have been taken within
             six months of study entry.

          4. Life expectancy of at least 12 weeks.

          5. World Health Organisation (WHO) performance status of 0-2 (Appendix 1).

          6. Haematological and biochemical indices within the ranges shown below. These
             measurements must be performed within one week before their first dose of mAb (BI-1206
             and/or rituximab) as part of this study.

             Laboratory Test Value required

             Haemoglobin (Hb) ≥9.0 g/dL (red cell support is permissible)

             Absolute neutrophil count (ANC) ≥1.0 x 10^9/L (or >0.5 x 10^9/L if due to lymphoma),
             granulocyte - colony stimulating factor (G-CSF) support is not permissible at
             screening

             Platelet count ≥50 x 10^9/L (or ≥30 x 10^9/L if due to malignant involvement of bone
             marrow)

             Either:

             Serum bilirubin ≤1.5 x upper limit of normal (ULN) unless raised due to Gilbert's
             syndrome in which case up to 3 x ULN is permissible.

             Or:

             Alanine amino-transferase (ALT) and /or aspartate amino-transferase (AST) ≤ 2.5 x ULN
             unless raised due to malignant hepatic involvement in which case up to 5 x ULN is
             permissible

             Either:

             Calculated creatinine clearance (Cockcroft Gault) ≥30 mL/min (uncorrected value)

             Or:

             Isotope clearance measurement ≥30 mL/min (corrected)

          7. 18 years or over.

          8. B-cell lymphoma patients only: patients has at least one measurable lesion by CT scan
             (defined as greater than 1.5 cm in one axis) or in the case of Waldenström's
             macroglobulinemia, disease must be assessable by the criteria stated in Appendix 6 of
             the protocol.

          9. Patients recruited to Arm 2 in Parts A and B (combination arms) only: CD20 positive
             malignancy as demonstrated by immunohistochemistry or flow cytometry prior to trial
             entry.

        Exclusion Criteria:

          1. Allogenic bone marrow transplant within 12 months prior to the first dose of BI-1206
             or presence of chronic graft versus host disease.

          2. Patients with clinically active leptomeningeal or central nervous system
             lymphoma/leukaemia.

          3. Doses of prednisolone >10 mg daily (or equipotent doses of other corticosteroids) are
             not permitted whilst on the study other than as pre-medication. During the screening
             period, doses of up to 20 mg per day may be given but the dose must be reduced to 10
             mg/day by Cycle 1 Day 1 (or Day -7 in the CLL combination expansion).

          4. Known or suspected hypersensitivity to study drugs.

          5. Cardiac or renal amyloid light-chain (AL) amyloidosis.

          6. Radiotherapy, endocrine therapy, immunotherapy, chemotherapy or investigational
             medicinal products during the previous 4 weeks before treatment.

          7. Ongoing toxic manifestations of previous treatments. Exceptions to this are alopecia
             or certain Grade 1 toxicities, which in the opinion of the Investigator and the
             Sponsor should not exclude the patient.

          8. Ability to become pregnant (or already pregnant or lactating). However, those female
             patients who have a negative serum or urine pregnancy test before enrolment and agree
             to use two forms of contraception (one highly effective form plus a barrier method)
             [oral, injected or implanted hormonal contraception and condom; intra-uterine device
             and condom; diaphragm with spermicidal gel and condom] or agree to sexual abstinence^4
             for four weeks before entering the trial, during the trial and for twelve months after
             completing treatment are considered eligible.

          9. Male patients with partners of child-bearing potential (unless they agree to take
             measures not to father children by using a barrier method of contraception [condom
             plus spermicide] or to sexual abstinence effective from the first administration of
             BI-1206 or rituximab on the study, throughout the trial and for twelve months
             afterwards. Men with partners of child-bearing potential must also be willing to
             ensure that their partner uses an effective method of contraception for the same
             duration for example, hormonal contraception, intrauterine device, diaphragm with
             spermicidal gel or sexual abstinence4). Men with pregnant or lactating partners should
             be advised to use barrier method contraception (e.g. condom plus spermicidal gel) to
             prevent exposure to the foetus or neonate.

         10. Major thoracic or abdominal surgery from which the patient has not yet recovered.

         11. At high medical risk because of non-malignant systemic disease including infection.

         12. Known to be serologically positive for Hepatitis B, Hepatitis C or Human
             Immunodeficiency Virus (HIV).

         13. Patients with an active, known or suspected autoimmune disease (not including CLL
             auto-immune disease). Patients with Type I diabetes mellitus, hypothyroidism only
             requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or
             alopecia) not requiring systemic treatment, or conditions not expected to recur in the
             absence of an external trigger will be permitted to participate.

         14. Concurrent congestive heart failure, prior history of class III/ IV cardiac disease
             (New York Heart Association [NYHA]), prior history of cardiac ischaemia or prior
             history of cardiac arrhythmia.

         15. Patients for whom rituximab is contraindicated due to severe previous hypersensitivity
             or any other reason (Arm 2 in Parts A and B [combination arms] only).

         16. Ongoing infection requiring treatment with antibiotics, antifungals or antivirals.
             Prophylactic use of antibiotics, antifungals or antivirals would not have excluded
             patients.

         17. Any other condition which in the Investigator's opinion would not make the patient a
             good candidate for the clinical trial.

         18. Is a participant or plans to participate in another interventional clinical study,
             whilst taking part in this Phase I/IIa study of BI-1206. Participation in an
             observational study would be acceptable.

         19. Current malignancies of other types, with the exception of adequately treated
             cone-biopsied in situ carcinoma of the cervix uteri and basal or squamous cell
             carcinoma of the skin. Cancer survivors, who have undergone potentially curative
             therapy for a prior malignancy, have no evidence of that disease for three years or
             more and are deemed at negligible risk for recurrence, are eligible for the trial.