Key Inclusion Criteria:
* Provision of signed and dated, written informed consent prior to any study specific procedures, sampling and analyses
* Japanese subjects at least 20 years of age at the time of study entry.
* Presence of radiographically measurable lymphadenopathy or extranodal lymphoid malignancy (defined as the presence of a ≥ 2.0 cm lesion as measured in the longest dimension by computerised tomography \[CT\] scan). Note: Not applicable to subjects with Chronic lymphocytic leukemia (CLL), Waldenström macroglobulinemia (WM)
* Eastern Co-operative Oncology Group (ECOG) Performance Status (PS) of ≤ 2
* Adequate organ function defined as follows: aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 2 × institutional upper limit of normal (ULN), total bilirubin ≤ 1.5 × ULN except in the case of subjects with documented Gilbert's disease, ≤ 2.5 × ULN
* Serum amylase ≤1.5 × ULN or serum lipase ≤1.5 × ULN \<Part2\>
* Relapsed or refractory CLL/SLL: Confirmed diagnosis of CLL/SLL, which has relapsed after, or been refractory to ≥ 1 prior therapy for CLL/SLL, and has active disease meeting IWCLL 2008 criteria (Hallek 2008).
* Relapse or refractory MCL: Confirmed diagnosis of MCL, which has relapsed after,or been refractory to ≥ 1 prior therapy for MCL, and documented failure to achieve at least PR with, or documented disease progression after, the most recent treatment regimen.
\<Part3\>
* Japanese subjects:
1. ≥ 65 years of age OR
2. ≥ 20 and \< 65 years of age, provided that they meet at least one of the following criteria:
i. Creatinine clearance 30 to 69 mL/min using the Cockcroft-Gault equation. ii. A score higher that 6 on the Cumulative Illness Rating Scale-Geriatric
* Diagnosis of CD20+ CLL that meets published diagnostic criteria (Hallek 2008):
1. Monoclonal B cells (either kappa or lambda light chain restricted) that are clonally co-expressing ≥ 1 B-cell marker (CD19, CD20, or CD23) and CD5.
2. Prolymphocytes may comprise ≤ 55% of blood lymphocytes.
3. Presence of ≥ 5000 μL B lymphocytes in the peripheral blood (at any point since diagnosis)
Key Exclusion Criteria:
* History of other invasive malignancy within 2 years except for cervical carcinoma in situ (CIS), non-melanomatous carcinoma of the skin or ductal carcinoma in situ (DCIS) of the breast that have been surgically cured.
* A life-threatening illness, medical condition or organ system dysfunction which, in the investigator's opinion, could compromise the subject's safety, interfere with the absorption or metabolism of acalabrutinib, or put the study outcomes at undue risk
* Significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure, or myocardial infarction within 6 months of screening, or any Class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification (New York Heart Association Functional Classification 1994).
* Malabsorption syndrome, disease significantly affecting GI function, resection of the stomach, extensive small bowel resection that is likely to affect absorption, symptomatic inflammatory bowel disease, partial or complete bowel obstruction, or gastric restrictions and bariatric surgery such as gastric bypass.
* Known CNS involvement by lymphoma/leukemia
* Known prolymphocytic leukemia or history of, or currently suspected, Richter's syndrome (for CLL/SLL)
* Receipt of any biological or immunological based therapies (including experimental therapies) for leukaemia or lymphoma or myeloma (including, but not limited to, MAb therapy such as rituximab, or cancer vaccine therapies) within 4 weeks prior to the first dose of acalabrutinib.
* Any prior therapy with BCR inhibitors (eg, BTK, PI3Kδ, or SYK inhibitors) or BCL-2 inhibitors (eg, venetoclax/ABT-199)
* Known history of HIV, serologic status reflecting active hepatitis B or C infection,or any uncontrolled active systemic infection.
* History of stroke or intracranial haemorrhage within 6 months prior to first dose of study drug.
* Ongoing, drug-induced liver injury, alcoholic liver disease, nonalcoholic steatohepatitis, primary biliary cirrhosis, ongoing extrahepatic obstruction caused by cholelithiasis, cirrhosis of the liver, or portal hypertension.
* History of or ongoing drug-induced pneumonitis
* Estimated creatinine clearance of \< 30 mL/min, calculated using the formula of Cockcroft and Gault \[(140-Age) • Mass (kg)/(72 • creatinine mg/dL); multiply by 0.85 if female\]
* Significant screening electrocardiogram (ECG) abnormalities including left bundle-branch block, 2nd degree AV block type II, 3rd-degree AV block, Grade ≥ 2 bradycardia, and the average QT interval corrected for heart rate (QTc) from the three screening ECGs must be \> 480 msec (calculated using Fridericia's formula: QT/RR0.33)
* Concurrent participation in another therapeutic clinical trial.
* History of bleeding diathesis (eg, hemophilia or von Willebrand disease)
* Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists (eg, phenprocoumon) within 7 days before first dose of study drug.
* Requires treatment with proton pump inhibitors (eg, omeprazole, esomeprazole, lansoprazole or rabeprazole). Subjects receiving proton pump inhibitors who switch to H2-receptor antagonists or antacids are eligible for enrollment to this study.
* Presence of a GI ulcer diagnosed by endoscopy within 3 months before screening.
* Uncontrolled autoimmune hemolytic anemia (AIHA) or immune thrombocytopenic purpura (ITP).
