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palindromes
A number is palindromic in base  $b$  (usually base 10) if the representation in that base is the same read from the right or from the left, as in 1257521, or in  $(1001)_2$  which is the representation of 9 in base 2.

A palindrome is nontrivial if it has more than one digit.

The first such numbers (in base 10) are 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of palindromic numbers

A graph displaying how many palindromic numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of palindromic numbers
Useful links Patrick De Geest, World of Numbers a website dedicated to palindromes
Mathworld, Palindromic Number
OEIS, Sequence A002113
Palindromic numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 101 181 + 967828769 ABA 242 3993 20402 + 8163248423618 aban 11 22 33 + 999000000999 abundant 66 88 222 + 49988994 Achilles 36963 2138312 2177712 + 40516277261504 admirable 66 88 222 + 8987898 alt.fact. 101 alternating 101 121 141 + 989898989 amenable 33 44 77 + 989999989 apocalyptic 222 434 646 + 29992 arithmetic 11 22 33 + 9999999 balanced p. 373 11411 30103 + 996989699 betrothed 5775 binomial 55 66 171 + 6874200024786 brilliant 121 323 737 + 999949999 c.decagonal 11 101 151 + 155075181570551 c.heptagonal 22 1331 1919191 + 38437311373483 c.nonagonal 55 595 5995 + 683727232727386 c.octagonal 121 10201 12321 + 900075181570009 c.pentagonal 141 181 10401 + 620873909378026 c.square 181 313 545 + 582818040818285 c.triangular 15151 45154 66466 + 479573060375974 cake 232 10701 canyon 101 202 212 + 987654323456789 Carmichael 101101 Carol 959 Chen 11 101 131 + 9981899 congruent 22 55 77 + 9999999 constructible 272 cube 343 1331 1030301 + 1334996994331 Cullen 161 Cunningham 33 99 101 + 995570353075599 Curzon 33 141 393 + 189959981 cyclic 11 33 77 + 9998999 D-number 33 111 141 + 7023207 d-powerful 262 333 373 + 9859589 de Polignac 373 757 959 + 99944999 decagonal 232 27972 76867 + 675972505279576 deceptive 7777 10001 11111 + 35300000353 deficient 11 22 33 + 9999999 dig.balanced 11 44 99 + 199505991 droll 4224 48384 Duffinian 55 77 111 + 9998999 eban 44 66 2002 + 66066066066066 economical 11 101 111 + 18800881 enlightened 119911 equidigital 11 101 111 + 18800881 eRAP 23444432 3686336863 98784948789 802959959208 esthetic 101 121 212 + 989898989898989 Eulerian 11 66 evil 33 66 77 + 1000000001 fibodiv 323 646 969 Fibonacci 55 Friedman 121 343 10201 + 825528 frugal 343 1331 10201 + 985383589 gapful 121 242 363 + 99990009999 Giuga 858 good prime 11 101 191 + 195353591 happy 44 262 313 + 9992999 Harshad 111 171 222 + 8998558998 heptagonal 55 616 3553 + 717685292586717 hex 919 1081801 1188811 + 131374494473131 hexagonal 66 3003 5995 + 636188414881636 hoax 22 202 424 + 98122189 Hogben 111 343 757 + 755971313179557 Honaker 131 16661 33533 + 982323289 house 434 28882 339585933 iban 11 22 44 + 777777 idoneal 22 33 88 232 inconsummate 161 272 383 + 969969 insolite 111 111111111 interprime 99 111 282 + 99755799 Jacobsthal 11 171 junction 101 111 202 + 94999949 Kaprekar 55 99 999 + 999999999999999 Lehmer 595 949 1111 + 153452254351 Lucas 11 167761 lucky 33 99 111 + 9986899 m-pointer 131121131 1116111116111 magic 111 505 5335 magnanimous 11 101 1001 modest 111 222 333 + 1999889991 Moran 111 171 222 + 89888898 Motzkin 323 mountain 121 131 141 + 234567898765432 nialpdrome 11 22 33 + 999999999999999 nonagonal 111 474 969 + 699030030996 nude 11 22 33 + 489939984 O'Halloran 44 oban 11 33 55 + 999 octagonal 8008 120232021 124060421 + 12212500521221 odious 11 22 44 + 999999999 Ormiston 1303031 1333331 1360631 + 977999779 palprime 11 101 131 + 999999787999999 pancake 11 22 121 + 706625414526607 panconsummate 11 77 121 353 pandigital 11 99 141 + 298989892 partition 11 22 77 101 pentagonal 22 1001 2882 + 264571020175462 pernicious 11 22 33 + 9998999 Perrin 22 plaindrome 11 22 33 + 999999999999999 Poulet 101101 129921 1837381 power 121 343 484 + 9420645460249 powerful 121 343 484 + 900075181570009 practical 66 88 252 + 8992998 prim.abundant 66 88 222 + 97922979 prime 11 101 131 + 99999199999 productive 22 66 616 2002 pronic 272 6006 289982 + 250087292780052 Proth 33 161 353 + 946783387649 pseudoperfect 66 88 222 + 999999 repdigit 11 22 33 + 999999999999999 repunit 111 121 343 + 755971313179557 Rhonda 512219912215 Ruth-Aaron 77 949 1331 + 804939939408 self 121 222 323 + 999939999 self-describing 22 4444 442244 + 88886666668888 semiprime 22 33 55 + 99977999 sliding 11 101 1001 + 100000000000001 Smith 22 121 202 + 99144199 Sophie Germain 11 131 191 + 999212999 sphenic 66 222 282 + 99988999 square 121 484 676 + 900075181570009 star 121 181 1441 + 396868131868693 straight-line 111 222 333 + 999999999999999 strobogrammatic 11 88 101 + 888888888888888 strong prime 11 101 191 + 9980899 subfactorial 44 super-d 131 181 333 + 9990999 tau 88 232 252 + 889949988 taxicab 4607064 2344344434432 109514040415901 tetrahedral 969 1771 6378736 triangular 55 66 171 + 684866959668486 tribonacci 44 trimorphic 99 999 9999 + 999999999999999 truncatable prime 313 353 373 + 799636997 twin 11 101 151 + 999454999 uban 11 22 33 + 99000099000099 Ulam 11 77 99 + 9999999 undulating 101 121 131 + 989898989898989 unprimeable 515 535 626 + 8997998 untouchable 88 262 292 + 890098 upside-down 55 555 5555 + 555555555555555 wasteful 22 33 44 + 9999999 weak prime 131 151 181 + 9989899 weakly prime 79856965897 91507670519 Woodall 191 323 383 99999999999 Zuckerman 11 111 212 + 2321111232 Zumkeller 66 88 222 + 89898 zygodrome 11 22 33 + 999999999999999