Inclusion Criteria:  Age ≥60 years Pathologically confirmed MCL (according to the 2016 WHO classification), with documentation of monoclonal B cells that have a chromosome translocation t(11;14)(q13;q32) and/or overexpress cyclin D1 Stage II-IV, measurable by imaging and requiring treatment in the opinion of the treating clinician No previous treatment for MCL (other than localised radiotherapy or 7-day pulse of steroids for symptom control) ECOG performance status 0 - 2 Absolute neutrophil count (ANC) > 1.0 x 109 and platelet count >100 x 109, unless related to lymphoma - in this situation, the threshold for inclusion is ANC > 0.5 x 109 and platelet count > 50 x 109 Creatinine clearance > 30 ml/min (Cockcroft-Gault) AST and/or ALT <3xULN and/or total bilirubin <3xULN Able to give voluntary written informed consent Woman of childbearing potential (WOCBP) who are sexually active must use highly effective methods of contraception during treatment and for 2 days after the last dose of acalabrutinib or for 12 months after last dose of rituximab, whichever is longer  Exclusion Criteria:  Patients considered fit enough to undergo autologous or allogeneic stem cell transplant for MCL Major surgery within two weeks prior to day 1 of cycle 1 Patients who are unable to swallow capsules, or who have disease significantly affecting gastrointestinal function that would limit oral absorption of medication Known serological positivity for HBV, HCV, HIV. Patients who are hepatitis B core antibody (anti-HBc) positive and who are surface antigen negative will need to have a negative polymerase chain reaction (PCR) result. Those who are hepatitis B surface antigen (HbsAg) positive or hepatitis B PCR positive will be excluded. Patients who are hepatitis C antibody positive will need to have a negative PCR result. Those who are hepatitis C PCR positive will be excluded Diagnosed with or treated for any other malignancy than MCL within 2 years prior to day 1 of cycle 1 (except basal cell carcinoma, cutaneous squamous cell carcinoma or any other in situ malignancy) Active infection requiring treatment Serious medical or psychiatric illness likely to interfere with participation in this clinical study Concurrent treatment with another investigational agent outside of this protocol Known history of drug-specific hypersensitivity or anaphylaxis to rituximab or acalabrutinib (including active product or excipient components). Active bleeding, history of bleeding diathesis (eg, hemophilia or von Willebrand disease) Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenic purpura) The use of strong CYP3A inhibitors within 1 week or strong CYP3A inducers within 3 weeks of the first dose of study drug is prohibited Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists (eg, phenprocoumon) within 7 days of first dose of study drug Prothrombin time/INR or aPTT (in the absence of Lupus anticoagulant) > 2x ULN Requires treatment with proton pump inhibitors (eg, omeprazole, esomeprazole, lansoprazole, dexlansoprazole, rabeprazole, or pantoprazole). Patients receiving proton pump inhibitors who switch to H2-receptor antagonists or antacids are eligible for enrollment to this study History of significant cerebrovascular disease or event, including stroke or intracranial hemorrhage, within 6 months before the first dose of study drug Breastfeeding or pregnant women Concurrent participation in another therapeutic clinical trial