NOVELTY - Vinyl addition norbornene-type polymer comprising a first type of repeating unit derived from a norbornene-type monomer (I) and a second type of repeating unit derived from a norbornene-type monomer (II), is new.    USE - The vinyl addition norbornene-type polymer is useful: for the hydroxide conducting anionic alkali exchange membranes and the active layer of alkali fuel cell (all claimed).    ADVANTAGE - The vinyl addition norbornene-type polymer: has low swelling in hot methanol and high ion exchange capacity; has improved polymer stability and stability under alkaline conditions; has improved mechanical strength without decreasing hydroxide ion conductivity; provides alkali fuel cell that are resistant to being blocked or destroyed by carbonate formation, thus exhibiting long-life with enhance performance; and provides alkali anion exchange membranes having high oxidation stability.    DETAILED DESCRIPTION - Vinyl addition norbornene-type polymer comprising a first type of repeating unit derived from a norbornene-type monomer of formula (I) and a second type of repeating unit derived from a norbornene-type monomer of formula (II), is new.    at least one of R9-R12 = functional group capable of quaternization, preferably the functional group is quaternized to comprise N,N,N-trimethylhydroxylammonium; and    others of R9-R12 = H, 1-5C alkyl, aryl or alkyl-aryl;    at least one of R13-R16 = maleimide-(1-4C alkyl) pendent group (optionally substituted) or another cross-linkable group; and    others of R13-R16 = H, 1-12C alkyl, alkyl-aryl or methyl glycol ether including (-CH2-(OCH2CH2)q-OCH3);    q = 1-4; and    m = 0-3.    INDEPENDENT CLAIMS are also included for:    (1) a ring-opening metathesis polymerization (ROMP) norbornene-type polymer comprising a first type of repeating unit derived from a norbornene-type monomer of formula (III), and a second type of repeating unit derived from a norbornene-type monomer of formula (IV);    (2) a hydroxide conducting anionic alkali exchange membrane (A1) comprising the vinyl addition polymer;    (3) a hydroxide conducting anionic alkali exchange membrane (A2) comprising the ROMP polymer; and    (4) an alkali fuel cell comprising either (i) the hydroxide conducting anionic alkali exchange membrane (A1) or (A2), (ii) a first electrode having an active layer, where the active layer comprises vinyl addition polymer or ROMP polymer, (iii) hydroxide conducting anionic alkali exchange membrane (A1) and a first electrode having an active layer including vinyl addition polymer, (iv) hydroxide conducting anionic alkali exchange membrane (A2) and a first electrode having an active layer having the vinyl addition polymer or (v) hydroxide conducting anionic alkali exchange membrane (A2) and a first electrode having an active layer including ROMP polymer.    at least one of R1-R4 = pendent group of formula (-R1a-Ar1-(-R2a-(N+)(R3a)(R3b)-R3c (OH)-)v); and    others of R1-R4 = H, 1-10C alkyl, aryl or alkyl-aryl group;    R1a = -(CH2)p-;    Ar1 = aromatic group having at least one aromatic ring; either    R2a = -(CH2)p- or -(CH2)s-O-(CH2)t-; or    R2a = coupled to N of quaternary ammonium functional group by a covalent bond;    R3a-R3c = CH3, aryl or 1-4C alkyl-aryl;    at least one of R5-R8 = maleimide-(1-6C alkyl) pendent group (optionally substituted), 5,5'-oxybis(methylene)bis(norbornenyl), 5,5'-(2,2'-oxybis(ethane-2,1-diyl)bis(oxy))bis(methylene)bis(norbornenyl), 1,12-di(norbornenyl)-2,5,8,11-tetraoxadodecanyl, bis(norbornenyl), 1,2-di(norbornenyl)ethyl, 1,4-di(norbornenyl)butyl, 1,6-di(norbornenyl)hexyl, 1,4-di(norbornenyl)phenyl or 1,4-bis(norbornenyl-methyl)phenyl; and    others of R5-R8 = H, 1-12C alkyl, terminally halogenated 1-12C alkyl, 1-12C alkyl-aryl (where aryl portion is optionally halogenated) or methyl glycol ether including (-CH2-(OCH2CH2)q-OCH3);    v = 1 or 2;    s, t = 1-6;    p = 0-12; and    x = 0-3.    Provided that cyclopentadiene and cyclooctene are not crosslinker moieties.