A honeycomb structure has grooves dented inwardly from the surfaces of the partition walls along a cell direction An open width of an open end of the groove is 0.015-0.505 mm and smaller than the open width a length of one side of each of the cells with the grooves, a bottom width of a bottom of the groove is 0.01-0.5 mm and smaller than the open width, a height from the bottom of the groove to the open end is 0.01-0.05 mm, a thickness of the partition wall in a groove portion is 50 m or more, a ratio of the number of the cells with the grooves to the number of the total cells is 80% or more, and a value obtained by subtracting the open frontal area when the grooves excluded from the open frontal area when the grooves included is 0.1-8.0%.

A honeycomb filter includes a honeycomb structure body having porous partition walls arranged to surround cells, in a cross section of the honeycomb structure body which is perpendicular to an extending direction of the cells, a value of a porosity of the partition wall in a partitioning region between the inflow cell and the outflow cell is defined as a porosity A, among intersecting portions where partitioning regions of the partition walls between the cells intersect one another, a value of a porosity of the partition wall in an intersecting portion between the two inflow cells is defined as a porosity B, and a value of A/B obtained by dividing the porosity A by the porosity B is from 0.50 to 0.95.

A honeycomb filter includes a plugged honeycomb structure body which has cell rows arranged along one direction in a cross section of the honeycomb structure body and including a first cell row constituted of inflow cells and a second cell row including outflow cells. A width P1 (mm) of the first cell row, a width P2 (mm) of the second cell row and a curvature radius R (m) of a curved shape of corner portions of a polygonal shape of each cell satisfy relations of Equations (1) and (2) below: Equation (1): 2100(P1/P2100)50, and Equation (2): 0.4(R/1000)/((P1+P2)/2)10020.

A honeycomb structure body has a skin part of a cylindrical shape and a honeycomb structural part formed with the skin part together in a monolithic body. The skin part and the honeycomb structural part have partition walls of a porous structure. The cells have first cells arranged adjacent to the skin part and second cells arranged adjacent to the first cells. The skin part, the first cells and the second cells form an outer peripheral section. A central section is arranged inside the outer peripheral section. The honeycomb structure body satisfies a relationship in which a thermal expansion coefficient of the outer peripheral section is greater than a thermal expansion coefficient of the central section.

A first circumferential wall disposed in a circumference of partition walls has no interface with the outermost circumference partition wall in a circumferential portion constituted by the partition walls whose wall thickness is larger than that of a central portion constituted by the partition walls in a central region. A maximum thickness of a total of the first circumferential wall and a second circumferential wall disposed to surround an outer side of the first circumferential wall is 1.2-3.0 mm, a difference between the maximum thickness and a minimum thickness of the total is 0.2-1.5 mm, and there is satisfied a relation, 0.5(TBTA)SB/SA100(%)9.0 in which TB and TA indicate average thicknesses (m) of the partition walls in the circumferential and central portion respectively, and SB and SA indicate areas (cm.sup.2) of the circumferential portion and the honeycomb structure in the cross section respectively.

A honeycomb structure includes a pillar-shaped honeycomb structure body having porous partition walls, wherein the partition walls have projecting portions, the projecting portions include specific projecting portions in which side surfaces have a divergent shape inclined at an inclination angle of 40 to 70, a relation of 0.04H/A0.40 is satisfied, a relation of 0.04H.sub.1/A0.38 is satisfied, the height H of the specific projecting portion and the height H.sub.1 of the hem satisfy a relation of H.sub.1<H, and a ratio of a total number of the specific projecting portions is 2.5% or more.

A honeycomb structure includes a pillar-shaped honeycomb structure body having porous partition walls surrounding a plurality of cells which extend from a first to a second end face and which form flow channels of a fluid, and a circumferential wall enclosing the partition walls, wherein, specific cell of the cells is provided with a pair of projection which projects into the cell from the partition wall, the projection projects, in at least a pair of opposed partition walls among the partition walls constituting the cell, into the cell from a first and a second side, and which is provided continuously in a direction the cell extends, and in the case where the width of a top part of the projection is denoted by W1, and the width of a bottom part of the projection is denoted by W2, W1>W2 holds, and W2/W1 is 0.5 to 0.9.

A honeycomb structure includes a pillar-shaped honeycomb structure body having porous partition walls, wherein the partition walls have projecting portions, a shape of each cell is polygonal, in the polygonal cell, the projecting portions are disposed on two sides extending from a corner of the cell, respectively, the projecting portions are non-uniformly arranged, a relation of Formula (1) is satisfied, and a ratio of a total number of the corners is 2.5% or more: Formula (1): 1/(N+1)<A/L.

A honeycomb filter includes a plugged honeycomb structure body which has cell rows arranged along one direction in a cross section of the honeycomb structure body and including a first cell row constituted of at least one of an inflow cell and an outflow cell, and a through-cell, and a second cell row including no through-cells. A width P1 (mm) of the first cell row, a width P2 (mm) of the second cell and a curvature radius R of a curved shape of corner portions of a polygonal shape of each cell satisfy relations of Equations (1) and (2) below: Equation (1): 2100(P1/P2100)50, and Equation (2): 0.4(R/1000)/((P1+P2)/2)10020.

A honeycomb structure body has an outer skin, cells arranged in an inside of the outer skin, and partition walls having pores. The partition walls are arranged in the inside of the outer skin. Each of the cells is surrounded by the partition walls. The pores have communicating pores which communicate with each other adjacent cells. Exhaust gas emitted from an engine passes adjacent cells through the communicating pores. The number of the communicating pores is at a density of not less than 18000 [pores/0.25 mm.sup.2] before a catalyst is supported in the pores. An exhaust gas purification filter having the honeycomb structure body with the catalyst is arranged in an exhaust gas pipe to purify exhaust gas containing PM emitted from an engine.